Salmon George
486
SALMON~ GEORGE A ~re~e on the ~TtCt~y~~c ~cowe~ry o/' three c~weyt~oyts Hodges and Smith
Transcript of Salmon George
SALMON~ GEORGE
A ~re~e on the
~TtCt~y~~c~cowe~ry o/'three c~weyt~oyts
Hodges and Smith
A TREATÎSEs\« &
OttTtttt
ANALYTICGEOMETRY
OP
THREE DIMENSIONS.
BT
GEORGE SAMÎON, D.D.,PBLMWAttD tMOR OP TtttNtTYOOLLEOB,ttOBHtr.
~Hx:
BODGBS, SMITH, & 00., GBAFTON STRBBT,
BOOKSNI.MRS TO THE TrNIVJEKSnT.
1862.
CAMBRIDGE:
t)H)tttB DY Vt!.].tt)t ttZTCt*)~ <HB)t)t <TZX!tT.
PREFACE.
IN writing a preface, what 1 am most tempted to
do is to enumerate and account for the omissions
of this treatise; if it were not that the size to
which the volume has swelled, renders it needless
for me to apologize for not having made it larger.
It may be right however to mention that the
chapters of this work were written and sent to press
at intervals as 1 found leisure, and that the earlier
part of the book has been in type more than a year.
This will explain why no use has been made of
some recent works and memoirs. In particular, 1
must express my regret that Hesse's "Lectures on
the Analytic Geometry of Space" came too late to
be of service to me.
In treating of the less modem parts of the Science,
1 have usually had Leroy's and Gregory'a Treatises
before me. The parts of this work which corre-
spond to the contents of theirs are, the Theory of
Surfaces of the Second Order, pp. 1–88 of the
Curvature of Sur&ces, pp. 197–883 of what 1 have
called the Non-Projective Properties of Curves of
Double Curvaturo, pp. 859–377 and of Families
iv PMFACE.
of Sttr&cett,pp. 813–338. Junior readers will pro-
bably nnd ail the information they require, if to
the course here marked out they add part of the
Theory of Confbcat Surfaces, pp. 139–138, and
the General Theory of Surfaces, Chap. x.
1 have to acknowledge with thanks the Mnd
readiness with which asauttance was afforded me
by any of my friends whose help 1 claimed. Those
to whom 1 am most indebted are Dr. Hart and
the Meaars.Roberts; but 1 have recelved occasional
assistance from Measrs. Townsend, Williamson, and
Gray, to the latter of whom 1 owe the list of Errata
which follows the Table of Contents.
1 have to thank the Board of Trinity College,
for their liberality in contributing to the expansé
of publication.
TamntCM.MM,Dcaux,j~y, 186~
CONTENTS.
--0--
CHAPTERI.THE rOtNT.
MMMethodofCo.ordinatea. 1
PMpett!jMofPto)Mt]OM 8Co-otdinateaof pointeutting in a given ratio the distancebetweentwo
pointa <Co-otdmxteeofcentre of<Het)'ahed)'oa aDiatancebetween twopointa (tectangtdaiCo-Otdtnatet) aDireetiom'coainetofaMne fAïM of a Sgm'ein tenna of MM*of ita pMJeot!oM .y r
Angle between twoUneoin terms of their dheetien.cottMt 8
PcrpetMMcalMdithmceof&point&onntHnB .8 8
Diteetiaa-cetineBaftheperpetidieulmtothephmeoftwoUnett 9T)t*!ffU'0!HfATM!tMCo-OMtMA'rM .9 9
DMtancebètweentwop«mta(oNiqaeCe-<ttdiMtee) 1t
DegreeofanequttiomuntdtettdbytnHufonMtion U
CHAPTERn.
ntTEBKtETATIONOFBQBATMNS.Meantngof a singleequation of a eytttm of two ot three équation* t2
Btery plane sectionof a Bui&co of the «t~ degree fa a eorve of the
dej~ee tt
Every right lino meeMa surfaceof the xO'degree in t potnt* 14Order of carre in epacedeNned 14Three ouï&oeaof degreesM,a, p, iaterMet fn moppointe t<
CyHaOiettmt&Mtdenned 16
CHAPTERUI.
TKE PLANB.
BveryeqMtionof thé arat degreetcpte<en«tp!me taEquation of a plane in terme of its direction.cosuMNand petpMMticutm
tromoy~in < ïaAxgtebetweentwoptMtef) tyrConditionthat twoptoneemay be MutatUy perpendioular t]rEquation ofplanein terme of mtetcepts made on axes 7r
vi COXTENT8.
MM
EquMionofptMMthtonghthteepoinM M
ïntNpMtationofteKneintMiequation 19Value of detentttnmt who*e eonttitaentt OM thé dtMcthm*eo<ineBof
thKefighttinet 19
Length ofpetpendtcuiMfroma given point on a givenplane 20
Co'ordiMtesofintOteotionofthteephmet 21Conditionthat four planesmay meet in a point .21Volume of tetrahedron in terms ofco-ordiMtee of ita vertiees 21
Volume oftetrahedton, the eoMttom of whose faeeaaïe given 22
Equation of tnt&ceepttMingthrough intersectionof given sur&ce~ 22
The equation of any plane can be expreMed in terms of those of four
givenplanes, 23
QctDtnM~iftn Co-o&Dnt~TBt .23
Anluu'!nonicMLtioof~bUfpI&nes 24'i'H)!Rt9KTt.ItK M
EqnntiotNofttrightUneinehtde&mcconBtantt 24
Condttiontbtttwotinetmayinto'sect M
Diïeetion'eotinetofttIinewhoseequ&tioMaTegiven 26
Equationaof peïpendieuJM:froma given point on a given plane 27Direetion'eotine~of the bMeotontofthe angle betweentwo Une* 28
Angtebetweentwoline* 28
ConditioMthat&Itnemay!ieinagiTenptane 28Number of eondttioMthat a tiae may Mein a givensurface 29Infereace as to the existenceof right tine*on mr&eet of the secondand
fhMdegreee 29
Equation of plane dnwn throngh a given line petpendiealar to a gtvenp!Me M
EqMtionofptaneparaMtotwogivenUne~ 30
Equatiom and length of shortest d!tt<meebetweentwo givenlines 8tPaOtZnTIBBOrTETittBBDRA S~Relationconnecting the mutual distancesof four pointsin a plane 33
VohnneefatetrahBdroninteonaofittedges 3!Relationconaecting mutualdistancer offour pointaona aphere 84
Radta*ofepherecitettmm[iMngatetrahedtom 81ShoMeMdistancebetween twooppositesidet St
Angteofindinatiomoftwcoppoeiteaidee 84
CHATTER IV.
KtOMM'tIMOFWADBIC9Ht GEt<E!tAt..
NumbefofeonditiontneeeMaTytodeteHnineaqMdtic 86
Re<MttoftMm!&mnationtopMaUdaxM 86
EqattiamoftangentphmettMtypoint 37
:Equation of polar plane. 86
Conetdt&ned–tMtgentcone 38
ïfocm ofhmmonic nM!MU)of radii through a point 39
livery hontogemeouaequation in z, y, repfeMntt a concwhose vertes
istheorigin 40
COMENTS. v!i
MM
DtMtiminantoftqMdrïe 4tCo-ordinttteoofcentte 4t
CondithmtthMaq<MdttetMyh&ve<mh'nnityefMntKe.. 4:
BqattthMtofcUmnetMiphme 48
Con)ugatedta<neteta 44
AqTMKb'iohMthMeptinc!paldtMMtTalptana& < 46
tormation of equatioumptesent!ng the three principal phn« 46
Beotaa~!e<under segmenta of intotcetimg ehor(t<propordunal te the
Kctm~e«mde)'theMgmenttofapt)trotpM<metehotdt 4~
EquatioM of tangent p]«ne and eone, &c., derived by jfoMMmBfhtI'"method .47
C<~tiMthattplMM'Mytouch<he6M'fMe 4C
CeNditionthtttttMMiMytonchtheMt&oe <
CHAPTER V.
CLAMITICA.'nCNOtf QUAMUCS.
RmctMm of eoefMents wMehare uMittted by KetNit~tm ttMMfbtma*
tion 6:
DiMrMnattngeaMc St
Cauchysproot that its roote amreal .M
BMpMidB M
HJPe1'boloidsof one and twosheets .M
Axymptottoconet .65y<a-aboM)ïft 1 M
ActtMireductionofeqMtienofapamboMd <9
CHAPTER VI.
OENTBAt.MM'ACES.
EqMttonK&n'edtotmot M
Lengthofaonnal 68
8um of aquateaof reciptoeth of thcee coojugate dimieteMla coattMt 62LoeuBof intetMction ofthree tangent planet whieh eut at right angles 63Sam of squares of threeeoajugate diameteraia <oMtMtt 64
PMaUelopipedoonatantwhose odget are oonjugatediMMtett 64Ataoanm of squares of ptojectiono of coi~ugMediametea, on tmy line
andoamypitno MLoeaft of inteifMethmof tangent pitmet at exttemMet of eo~ngate
dumeteK 66Quadratie which detMmimealengthe of axes of a central section 66
when the qmtdnc is ~t~enby the general equation 6?CtMtILAttSBMtONa 68
TonnofequatiomofeoMycUcmtt&eea .69Two ciMalat ~eotioMof oppositeayttenMlie on the eatnetpheM MHImMUcsdenned ?!Circulor MotioMof paraboloida fil
viit CONTENTS.
tJMt
RECMMMtABGBItMATOM MTwolines ofoppositetyttMMnuMtinteraeet 73
NotwottMtoft'hesatMtyetcmintersect .74Distinctionbet~eendevetopabteamdthewMtt&M& 76A right line whose motion is Mgaiated by <~t* conditionsgeMMteea
surface M
Surface gcneratedbyaline meetingthKcditeetorMneo 77
Itight Unesonhyperbolleparaboloid ?9Four gencratorsofona systemeut anygeneratorof theother in a eonttMt
anh&nnoniertttio 80Surfacegeneratedby lines joming cotreapondingpointa on two homo-
gtttphicailydMdedtinM .80Conditionsfor aui&ce<of tevohtUon 82ExAMTI.BaOF1.001 .84
CRAPTER Ttl.
METBOD80F ABMDGE))NOTATION.RedprocalofacnrfeinspMe 89
OtMlfLt!ng'pttneof&CNrMde8n9d 89
Degreeofthereetpmcalefaquadtic 90
Equation ofsystemof quadricahaving a eomaMMteurve 90A)l quaddeBthrough sevenpoints pM$through an eighth 91Locm of centres ofquadtict through eight point*ot touching eight planes 99Four conespfes through the intersection oftwoqa<~d~~<!< ..93Conditionthat two quadriesmay touch .94The point of contact of two Mt&ces whichtouch is a double point on
theirinteMMtien M
8tationarycon«n!tdenned MThree quadrics having a eommoneurve cm be dMertbed to touch a
given plane, and two to touch a giveuMne .96Thé pointa of contactwith a given plane of three coneycUeaut&ceosub.
tend right angleaat the centre .9?Casewhere two quadnM hâvedouble contaot .97
SimilarqnadriM 98Gtometneal tohttton of ptoNemt of eircular MtMont 98Foct 99TwoHndaoffoci .t00Foealconice 101
Analysisof spodesoffocal conioafO'eachMndofqnatMc 102T'oeaUmeBottKxme 104toeal !inet perpendicular<oeircnlar sections ofreciprocaleone 106The aeeRontofrceiprocalconeabyany plane NfepoiarKeipioeala t06
ToealeoniMofpaTaboloida t07
B'ocMand,diieetcixpMpetty6f9mrfa<)M .107The tangent cone,whosevertex is a focua, isa right cône 109
Quadn<Mtoochinga)ongaphnp''urve HO
COSTENM. ixMM
Propertlesof theif sections .111Twoqnadïiea envelopedby the tamathird imteKeetin plane conrM 111Fennof equation referred to a eeH'-conjugatotetrahedMn.. Il l
Propertiesof invariantaof a «yMemof two quadrio ..112PMte'a ptopetty ofephetet c!ttnnMCttbinga teU-conjugatetetrahedton 113
Beme'6prope)tyofTe)rtice<oftwoBeM-eonjugatotetrahedTa 1133Chasles' ptcpetty of Bnea jointe ecmeapondingvetUceaof twoeo!~n-
gatetetmhedrtt .118AaatoguMtoPMtal'tTheoMm tM
Equation of sphore ohcumtenMng a tetithedron, (for iMt~bed epheM,aeep.lM) n?
Equation oftHpherefntetrithedtfJco.ordtnates 118
Twoq<M~Me)KMYa)'iAotto<tp)ntcfq<MtdHM 119Bqufttlo&repreaenting the facosof the tetrahedron Mlf-cet~jugittewith
regard to two qmtdncs 119
Equation of developableoiroumscribingtwoquadriee 1MConditionthat a line ahould meetintettection of twoquadrica 121
Equation of developable generatedby tangents to intenection of two
quadrics .< tMUnes in whieh the developablemeetseither surface t28RBCIpmoOAiSfNJ'ACEB 1M
Reciprocalof a qaadtio when a Mt&eeof revolution 124
ReciprocalofanJedMirfaceisaintedtnrfaceoftheMmeaegtM t24
l'roperty of umbi1!carfoci obtainedby rociprocatiOD 129
CHAPTER VIH.
CONMCALBPRFJtCtN.
FoeatconteitheHmiteofoon&teatMAect tSOThrec con6xa!: through a point all real and of dH&tent<pec!eB t91
Two confooa!*eut at right angles 18SAxes of centralsection in-terms of axes of confoea!<thKmgh extremlty
oftOt~ugatediMneter 131
~DcoMtantatongtheinteMMtionoftwoconfoeab .136LoenBofpote of&xedpTanewithMgaTdto a systemofeonfoeals 136Axes of tancent coneare the thrcenonnehthtonghita vertes 137Transformationof equation of tangent eone to the three normaleas axes
otco-otd!nates 139ConeschremMcrtMn~confoctt!surfacesare confocal ItttThe focal lines of these cône))are the geneMton of the hyperboloid
thronghthe\ettex ItS
Bectptoc<~ofeottfbea!tMeeoneyeUc 113
Tangent plane*through any I!ne to the two confoealswhich it tonchee
tttenmtuaUyperptndteuItt 1MTwo con&teahseen from any potnt appear to eut at right angles 144Nonnah to tangent planea through a giron KM generatea hyperbolie
paraboloid .141
CONTENTS.x
MMChtfUet'methedof obt~niBj;eqMthmof tangent coae 146
Intmeept on a bifocal thetd between tangent plane and pt«tM phnaHttca~hcentM .M8
GiveathMecon})<gatedtmMttMoftqQ<d~eteandi)M<aM5 M8
LoeMof~ftieMofrightconeBeavdophtgtMT&ce. 149
I~eM<)f!&tMK<<~ihtMmat)MNypMpen<UeM!<))'tangentltM< 1M
Ccneipo!t4mgpoin<aene<m&«!ab 1M
ItMy'ttheoremMtothedittaneeoftwoMchpointt 1M~McM'tanalogue to the phme theotem that the eam of foed dittOMee
i«!OMtMtt 1MLocusof pointaofeontMtof pNMtUetp!<mMto~eMnga <ettM«ifeM&caIa 1M
C<m&<~Mein~bed!nte<mtm<mdeTet()p<tNe .ÏMCmtYM'MttOBQu&DMCt 1M
B<dRofe<ttvatttfeof<HMnnalandof<tnobMqae<ee<ien tN)HneofoarMtutede&Md 1M
CoMtfnûtîonibrptMcdp~centFM'ofcutv<tttM'9 .161
8at&Mof<!entK<t:tt:eqaatMtnhow&and Ml
ÏtsMet!oM~ptinciptlphmea .1MB<}Mti<mofitBieciptoc<d 1M
CHAPTERIX.
CONBSAND6PKEBO-CONICS.
SphetMM-OtdiMttM .IM
CyeUc<t~ofspheM-<!<miea<malo(~u<teMy)npto<e)! 1698amcî<bct)lditt«mcaK!OMt<tnt .170
Fooatanddireettitpropertyefttphero-ccttiea HTZDi&teMe ofeqtMtMeofaxM of centMlsection of a qM(Mo,proportional
topMduetofNMeofaDg!MtmMts:wttheyd!ethmet 179
~a~onof~h<MintctU)ediaatetrahe<tMn 176
Eqmth)toftt4j;htoo)M 1M
Ofi<ghteoMtgtventhMeedgea,M'thteetangentphtnM 177
CHAFTERX.
ORNEBALTBEOBTOP8UBMOM.NtmbMoftemMingenentlequttNMt .179Seothmof eat&ee by tangent plane bas point of contact &<t<(doaNe
point 180
AtM&ceineenetitthmMptetangentphmee 181
t Idexional taDgenta de1Ined 1M
TheM]ettih;~pti~h7pet~M~~paMMie]Min<B 1M
Co~~tet<mgen<< 1M
TtttgeMptaM<ttapaMMiepoimtiaa<h)ab!etm~tntpIane 1M
DeaMepcimtBcnatKMtMe 186
AppBattionofJoMhimsthfd'tmetM .H7
CONTENTS. xi
M<H<Nttmhe)!of double tangent !!ntt which can be OHtwnthïMtjjha point on
<mot6Me .199
B'<M-m<ttt<mofequttic&ofttmgente<metoamt&tce 190
Numbet of !a&<x!oMlor double taagonte which con be drawn thronghany point .191
DegMeofïeeiptoe~MïtMe. 1M
Di.MtnainantoftHŒttMe.19!
Fo~qM~oofapMttMtepainttttMM 1M
HettttmoftmatfMe .1M
Nu!nbetofet&tioMïytan~e!ttpl<mMwMehptMthMOghtpoint M<
Evetyt~ht!ineon&t<tï&cetoaehe<&eHMdMt .199Cc&f~TonBMS~MACM 1MB.tKUmK'fentiratateofMinnalMCtion 198Ect!et'9<otmnta 199
Mem~iet'etheotem .Ml
Twot.pheMBh&tettttioaMyeontMt ÏM
~<daetofpttMipalï!t<m«tanypo!nt .S!M
I~ctMO<'po!n<twheM]ra<N.areeq~MlaaA<)ppe~te Mit
B)~M]~Mgd~M~N]~~Ù~~M ..20~CondttiOM&rannmMlio M6
HmetotspheriBalMtrv&tare .i!07NmmbeTofumbUt<aonttMttMe<t<')tth<M'<k)[ 208
St&tiot~eont~~ptieecontMtattwûpotnte .MDetemination of monneitwhichmeeta contecuNvenoKMl 209
BetttMd'tthe~otem-vaMre .0Uneeoîearvatmfe 211
ThehdMfeîentMeq'Mtton .M
I~e<)<)teaM<ttaMofe!ltp<i<M :lt
Dnpin'e theorem .t4Ift~o Mï&ces eut at right angles, their !nteMee&)n,if a lineofcMrra-
tareonone.isMontheothtt 216
LoemofcentKaalon~a line ot e).a'v*taKMa cuBp!dttle<)ge<m<nr&ceofnorma1e .M?
PtepertteBoteutfiMeotcentrett 917rOeodMieUneade&Mtd.S!8
Ktl~ofenrMtaMtepl)methm)tt[eet<HMt<mtNtgbwithtmgeatp!<ne !MLenetefe theoremof variationof anglebetween tangent plane and oseM-
lattttj;pl<meofMDeofeatv<tm'e 222
AgM(tetlclineofcarMtuMnta))tbeptme .2M
CHAPTER XL
CCMESANDDEVELMABLM.
8M. I. PtMBMt~ rMMMM ~4
Diïecti<m-oft<mgMttoaeuTve XMThtOfyofdeTebpftNeaMpt'~aed ï~
xu COXTENTS.
Equation ofepttnewhoMtqnMicnMntaiMOMpanMMte)' 228ChameMtiMiM .M9
CaspidaîedgeoftdeyelepaMe .Ma
Stationarypointtandplanea MS
Caytey'teqnaticm connecting tingaMRee of a curve m tpaea 2S4
Developablegenetatedby tangent* i<ofoamedegreeMteciptotat deve!op-able .296
8BC.tï. Ct.AM)KC*T!ONOPCcttTtt 240
Lo<iUtofvettexofquadtice<met)u<Mgh<mofMvenpointa MtA twittedenbieeanbedMcribedthM~htix points M
PmjettienofatwhtettwMchMttdeaMepoint .243PMpertieeoftwiMMicuMca. 244
'nMtT<MS9iaat<peeiet.48
Singi)Mtte«tfearveofintetMetion<~t'!MMtfiKM Mt
Numberof apparent doublepoint* ofintemeetion .MOCaoeofent&ceBwMehtonch Mt
EqnMioat cMmeetht~BhignMt!eB of earMa which together make upinteraectioa of two sur6acee M2
Tarodiatinct familiesof quartics 268
Fonrquartias offécond&milytht<Mt~d([htpointt !tS7Commoncarre on three aarfaoea equil~tent to how many pointa ef
intersection. 268
SinguMtiM of a doubleeatve connectedwith those of ita comptementMy 269SEC.in. Nott.PMJMTtM PROfBtTtMor CmtTM M9
Ï))jM!tio)t-co!iaei<fnonB<tlpltme Mt
Equation ofetcuMagptme MtTheheux :M
Equation ofoMmiatingptMeofinteHeetionoftwoMrfMM 26SConditionthat fourpoint*may lie in a ptane !M6
Radi<MofabMt<t<«mdot<)phenoalentvtttnre M9
ExptemioM&rangteofeontact 267Radiusofeurvataieofintenteet!onoftwoMriaeeB M9
Expressionforangleof torsion 269
Otodatingfighteene .270
Reot!~ingdeve!opaMe 271
Bect:f}'int;tnrfaoela surfaceof centfMet original developable 27)!
AnglebetweentwoMeccMiTeraduofeM'Yatttrc Z72
CaspHaledgeofpehtTderetopaMei< loca)ot centres of aphettcat cmrva-tore .274
BveryOMrehaaaninBnityofewtutet 274
TheMaregcodettMontnepotardeveIopaMe 27<
RadMteotaphM'ethumgh~arcoMeenttve pointa 276Co-ordinateaofitocentM .Z76Hietoryoftheotyofnon-ptanecnrvet 277Bao. IV. CcnYEtMtteBDexScMfAcat 278
DimistentMieqatttianotogeodeme 279Une joining Mtremttiea of indennitety ntM and equal geodedeacntt
thematnghtmgio) .279
MM
COXTEKT8. X)U
MM
RttMutofgeod~iceurvttmt .MO
pD constant for a geodeeteon <tqu<(dric .MlValueofthe coMtott thé same for aUgeode~Mthrough an umbUto 284Mt. M. Rabota'dedaotioMfKtmtMttheotem M6
IA)uviUe'attMM&rmati<moftquatt<xtpD'=constant 2MChMiet*ptoo&of this theorem .288
te aattMtenetonaaftt M9
EUiptMCO-<M<UMte< !?<
AteaofitatfftceofeNp~oM M2
SeeondiMegTtlof equation of geodeaia 292
lengthofegeodMie 292
GeodMiopotMeo-MdhMtee !93
Dj'.Hmt'BptoofofMr.RebeMs'BexpteMiom M6
UmbUic~tgeodMiMdonotKtninonthetMd~ee 800
lànes of level 800
IjBetotgteMeetsIope .301
GMM'ttheoryofcntv&ttMeofs'ja'BMM 802
MeMateofcurwtutauMttetedby deformation 804
To~cnrMtuMofgMdetiotritngteenanymïf'Mie .310
CHAPTERXYÏ.
FAMmOMOP SCBrACBS.
Eqwttt«MinYolting<t~ngt~<)rMttMyfttno6en 813
CyBnMctl MU'&eM .916CoaicatMt&CM .916CoMÎdtdMT&ces 318Sttt&uiMofte~&ticn 820
Oidetcf<tMM9BtM~Mtimof&&mi!yimMlvb!g~<Nn<)tiMM 3M
Sta&eMgeneMtedbyUnetpta'aUOttot~edphme 326
OrbyUneewhiBtnM9ta<b:ed<mie .328
Di&ren<i<leq'MttoMof)mted<m&<it< .980
TheMryofenyetope)t 331
DetcnBt~Mtïotiof arMiNryûinctiûM 333
pmtifddi&MntMeqnathmcfdeMhtptNex M8
'BtetrPM-HeMtm 338
TaMMMt&<!M .338
MStKntMequzthmofehaMetetiMhM ?9
Di&rentM equation 0~6~ on
BnMDSmtPACM .346
NtttoteefoontaetalangtmygenetatM 3«
Donble eittvet geneMNyexitt on mttd tW&ees 3M
8tcr&eeegener<ttedby<tIinetBït!ng<mthjee&xedditeetCM 3!0
NotmtttstdongagenttatorgcneretepaMMoid .351Linos of striction .354
xiv CONTENTS.
CHATTER Xïn.
SUBFACBfDSBIVEDMOMQCADMCS.MM
Bq'M&mofwMMMt&ee 388
IteeeotIeMbyptimipalptMtet .MfAptHetmr&eeB Ma
ï'<datTec!pMealûftp<!cM,ttptfdalofteciptM<d 3M
DëgTeeofMdpfoeatofwtYefmr&ce MO
Cteomet<le«tinvestigationof planeswMch touch aïong eifch's 36!!
Eqa<tttontneNiptieeo-<H'aMMttM M2
R<~tM~onfimaBg!.ebttw<e&ttatgentpltM<fndM<)iueveot<~ !M
Cbm~tm~kmfottM)~tphtne&tanyp<)iBt 368
Li)tM<tteatwt<)M<)fw<tT9MN&tM .MB
SM&cepaMtMtoempMM. 389
The<~<~dedv~p<!Mmt&ce< .370
Ih'opattlmofin'MKemt&eet Wt
L~of<mMtaK<<fnr&eeot<i!aMt<)!ty 37S
MMtmep~TepsMctttqMtdtie 9M
PtoNemof&idhtt!MeM~'Mpeda!aM6nt!c~Wtththat ofan~BgpttmNelMt&eee .?6
CHAPTERXIV.
SUttrACESOPTHIM MMHLEB.
CaMeehtfnnftdoubleMneB ~7
CnMe<)h<ni.n:doTtHepoimts S!9
Sylvetter'scanonieat~iam&teqwattMiofcnMe 8H
CMtMpo<td~p<~at<entheHeMian. 8!)2
R~t!ca<~th<<a~ephnet)totheHM<iM)L M
ToI«teuMexoftpItme Mt'Btete~tMth~MHM~n 886
B~~tHnoKmotMct 388
Nwntbet~tdptetMtf~ntphmM M7
SohSCi'sMheme&trthetwenty-MvealitLea .3M
]&).volut!<mnftixHneo!nepMe 890
Condition th~t ave Mnmshoatdbe met by a eemmnnttaMveKOl 390
AzàtytiBofeptciMofeaMM S9t
hv~ntttmdco~no-tanttofeaMea 3M
t~m~M~M~M~~&M~~ 394
Ttve&nMtMnentalin~tiamM .39~
EqMtt!en<)fMt6!MMwMchdetet)mhMStwMtty~te~enUn~e <<?
CHAPTERXV.
GBtEML THEOttT OPSMtFACEa.
Degree ofeen~en that three Mx&eeamay havea eMmaon tangent lineatthdtintetMctton Mt
Degteeofeemdttionth~twotnn~MmaytMeh <M
CONTENTS. XV
MM
Ot4erofjeve)cp<tHeenvetoptngaMr&te<'lone<tghMmeaM6 404
OfdevdcptNegeMMtedbyaMnemeetia~tiM~vencm'VM <M
CoN'ttetOttnnNWtTKtt~BFACM M*
Lo<'Mofpo!atsofoenttM!tofdoab!e!nae~oMlt«nge&<< «?
Andeftttptetangento <M
Centttctcf~bmteBwithsat&eef) .409LMM<~po!nttof<~taet<~dou1~tange~pI<me< *M
TtCMMatBRtM'Mcju.atnMAtM .<M
Nnm1)M~~p~tm~~p)<mettotMT&ee <t?
Ea~~mntMpkthtMOttttegreeoftM~pMeal~t
AM~~m~dM~~NM~&M~~M~N~ <M
Appt!cath)nt6t<t!td<na[&eM .?6
APPENDIX.
OnQMatemi<M)B
Omt)'tp!eo)ftho)~)!mlttyst<tM W
0&CtebMh'BMdcatat!<moî)MtfMeS. «*
Onth''rTf!e'-«f))y<tenmofeqnt).<i<NM) <M
ERRATA.
Ftgû LtM
S, C,J%f C, D, 8, rend A. B; C.
47, 4,~f drawn, feo<<be drawn.
80, 6, for plane of, «ad "ptxne, if.
9t, 4, for etghth &xed point, *Mtt eight Cxed pointa.112, IMt line, third group, for e' <WKf<tV.
148,. tti, the thecum hwre aoenbed to JecoM, had been previoM!y puT)'Ut&<!t[by Chasles, ~t'OMOtSt,xi. M).
189, M,M«d\19! 10, for <y, f«~ 61.
t99, e,j~X)t(«-!)(tt-.2),M<t<i'<t(<t-t)(n-).
J99,t<R.
201, 8, e~«' D.M, «Mtt< CM < <t~' jHt', <<tMf<CM*'t'.202, The determinant at mot of page onght to be bordered horizon-
tally and TerticaUv with .M, JV.
209, M,/M'a-~Ma<<<<. X
atO, 4,iC,tw<t~2t0, 4, far ~d, ~C, rtad 2A' 2a'·
210, 6 ofnote,a<ter2<<HM<l.216. t3,~MO.t«x<!70.SM, 9,~f278,yM<<:80.Mi, 9,Afi<,<we<<whBn.3:6, M, and 249, Une 7 &o)mhottom, for dmtbte points, <'««<double edg<e.MO, 6, In the values for 2g and 2i)',a &etor is omttted.
2y =<f<'{f"' (3f< + &e., Se =*ftf {ftf (ft + &c.
SM, last lino, for "princtpfd," read "commonconjug<te."M4, 2, P Md mean tangent planes to U and Yat the point.M4, 17, for "meets the quaddee," f~ Ifmeets the quartic."
266, tmt line but two, distant," fw<< distinct."
269, t4,~4m,fM<<aM.263, t6, trammose y'e and t~.267, Note,~)-CC,Ma<<~B.
B
ANALYTICGEOMETRYOFTHREEDIMENSIONS.
CHAPTERI.
THEPOINT.
1. WE dave seen already how tho positionof a point 0in a plane !s detennined, by referring it to two co-ordmateaxes 0~ 0 Y drawnin the plane. To determinethe positionof any point P in space,we have only to add t&onr appamtnaa third axis OZ not in the plane (sse SgnM next page).Then if we tmew the dMtamceof the point P from the plane
XOY, measured parallel to the line OZ,and aho iknew the? and y co-ordmateaof the point C, whare~C parallel to OZ
meets the plane, it is obvious that thé position of P wouldbe completelydetermined.
Thus, if weweregiven the threeeqt]at!ona<B=<t,y!=t,<!=:c,the nrat two equationawould determinethe point C, and then
drawing throngh that point a parallel to and taking onit a length jPC=c, we abouldhave the point
We have eeen aîready how a change in the aigu of aor b affects thé position of the point C. The agn of e will
determine on which side of the plane XOY the line PC isto be meaaured. If it be settled that lines on one aide ofthe plane are to be considorodaa positive,then those in theother d!rect!onmnst be conmderedas negative. Thaa, if wa
concelvethe plane Jï'<?y to be horizontal, it ia cnatomary to
THE pOINT.2
consider aa poaitive the a of every point <!&!<?that plane,in wMchcaae the e of every point &e&Moit m<Mtbe ooantedas negaëve. It M obv!oa9 that every point on the plane haaita<!=0.
The angles between the axes may be any whatever; butthe axes are said to be reetangnisr when the lines O.Ï, <?yare at right anglesto eaeh other, and the UneOZ pMpendicntar
tothephnejroy.
2. We have stated the methodof representing a point in
space, in the manner which seemedmost aunpie for readers
a!ready acquainted with Plane Analytic Geometry. We pro-ceed now to state the aame more BymmetncaUy.Car appa-ratoaeYMknttyconsistaof ~ee co'ordim&teaxes
0~,OY, OZmeetingin a point 0, which,asin Plane Geometry, !9ca]M the origin. The
three axesare caUedthe
axe8ofa!,y,erespec-ttvely. TheMthMeatBMd~nmineaboihreeco-
oïdinatej!&MM<,namely,thé ~MMsjcor; r<M;
wMohwesha!!
eaU the planes aw, M,jMcïeepect~ety. NowNBeettiBplamth&tJ~=!C!JB''=o,PB=CP=~, we may say that the poat!on of any point P
NtmMmK~~mMg~mïh~MeM~~m~M;~&dwm
parallel to the axis of to meet the planeye, PB parallelto
the <udaof~ to meet the plane «fc,and PO drawn parallel to
the axis ofeto meet the plane a~.Again, mneeO.P=at, OJE~ OF=e, the point given by
thé equations!c='<x,y~~ <!=c may be found by thé Mlow-
ing aymmetncd eomttmetion:meM~o~on the axia of !?, the
length OD'=ot,and through D draw the plane PBCD paraUelto the planey< meaMtteon the axisof yt OjE'=&,BnAthrough
THEPMNT. 8
B2
Fdraw the p!aneA1C!F paraUelto n: meamre on thé axisof < M'*==c, and through F draw the plane F~JÏF pataBdto «y: thé intemectionof thé three planes ao drawn ia the
point P, whose constntcëonis Mqa!re(L
8. The points C, D, E, are called the ~o~tc~M of the
point Pon Aethtee co-ordimateplanes; and when the axes are
rectangdar they are its o~<~M)o<projections. In what fol-
lowawe shall be almost exclusivelyconcernedwithorthogonat
protections,and thereforewhenwe speatcaimplyof prq}ect!oaa,are to be onderstood to mean orthogonal projectiona,unlesa
the contrary tBatated. There are somepMpertteaof orthogonalpro}ectMMMwhich we shali o~en have occaaïonto employ,andwhich we theroforo collect here, though we have gtven the
proofof some ofthem aiready. (See C~MMs,p. 81&.)3'~e<Mt~ of & oy<~oK<t!jM~!e(&<ao/'ot~Me )'~< ??
on <!<t~plaM M equal to tbe ?« tMt~KM? by <&!COSMteofthea~!6* ?&«!& makes<0~ thep&tKe.
The angle a !ine makes wMt a plane is meamred by the anglewhich the Une makes with its orthogonal projection on that plane.
The angle between two planes h meaeared by the angle between the
po-penNeutaM drawn in each plane to theh line of intmaeet!on at any
point of it. It may aho be meMMed by the angle between the perpen.dic<thtMlet &U on thé planes &om My point
The angle between two HnM which do not Intemeet, h) meMMedbythe angle between paMUebto both dtawn through any po!nt.
When we apeak of thé angle between two lines, tt h <M)-aMeto e]qn'e«twithontambi~ty whether we mean the acnte or the obtMe angle whieh
they make with euh other. When therefore we <peak of the angle be-
tween two Knee (&timttMee PP, CC'in the n~ne, next page), we ahan
tmdeMhmdth&t theM !!neaaMmeMmMdint!M<MMe<<Mt&MnJPto~andfrom C to < and that the parallel PQ h mea<UKdin the Mme diteethm.
The angle then between theae lines le aonte. But if we apokeof the anglebetween J*~ and C'C, we ahould draw the paraitet PQ' in the oppositedirection, and ahould wïah to expteM the obtuse angle made by the
«htea with each other.
When we fpeak of the angle made by any line OF with the axes, we
ahan always mean thé angle between OP and the jp«M<MMdireetton ofthe axes, VMhOZ, OY, OZ.
4 THEPOtNT.
Let PC, .PC" be drawn perpetMHcNhufto thé plane ~<?r;and CC' is the orthogonalpro-jectîom of the line PP on that
plane. Completethe rectangleby drawingPQ parallel to CC',and PQ will aiao be equal to
CC'. BntP~=.JRP'coeF'P~.
4. ï%e projection on anyplane of any <M'eatM another
F&MMM ~MO?<? tlte <M'~t)Mt~area mM&tpKM?by Meco8ineo~de angle SettceeKthe j)&tKM.(See C~t<M,p. 81S.)
For if ordinatesof both figuresbo drawn perpendiculartothe intersection of the two planes, then, by thé last article,
every ordinateof the projectionMequal to the correspondingordin&teof the original figuremultipliedby the cosine of the
angle betweenthe planes. But it was proved (<XM)MN,p. 29S,)that when two Sgorea are Ntdt that the ordinates corres-
poudingto equal ahscissc have to each other a constantratio,then the areas of the figureshaveto each other the sameratio.
S. The projectionof a point on any ??) is the point where
the line is met hy a plane drawnthrough the point perpendicnhrto the Une. Thus, in figure, p. 2, if the axes be rectangalar,
D, JE,F are the projectionsofthe pointP on the three axes.
T~e~O~CfMKof 0 ~tt<e right ?? ~XMt<M!0«5ef lineM eg<M~<0 Me~< ?? )Mt~~M~ Me C<M<<M<&e<M~&&e<tMOt<i~%M<t.
Let PP he the given line, and DD' ite projectionon OJF.
Throngh P dtaw P~ parallei to
OZto meet theplanejP'C'.D'; and
amee it is perpendicular to this
phne,the angle.P~jP' ts right, andFQ=~P'CMF'~<?. Bat~and.M)*are equal, ance they are the
intercepte made by two parallelptanes on twoparallel right !mea.
TUBMMNT. 6
6. If <j5e~ any threepoints P, f, P", the pro~M ofPP" on any line <e<K&eequal to the <MMof <~ pM~ec<«MMon
~<!<&fM<2/'PP'<!M<fJP'F".Let the projections of the three points be jy, D", then
if 1Ylie betweenD MtdD", DD" is evidently the oam ofJDJ~
and 2yD". If D" lie between D <mdD', DD" Mthe <?~eaceof J92)' and .D'.P"; bnt since the direction from 2)' to D" M
the oppositeof that from D to D', DU' is still the a!gebr<ncsum of Z)D' and 2yD". It may be otherwiseseen that the
projection of F'JP" is in the latter case to be taken with a
negative sign from thé consideration that in this case the
length of the projection!a found by m~tiplying P'P" by thecosineof an obtuseangle (sec note,p. 8). In general, if theMbe any number of points P, JP*,JP",P", &c., the projectionof PP" on any line is equal to thé Munof the projectiomof
PP', jFF", P"F"
7. We shaB have constant occasionto make use of the
followingparticular caseof the preceding.the co-o~MM<M any point P 4ejpM~c~ on any line,
Me<M)Mof the threejM~eetMMMMequal<o ~w~c~MMo/'the
radius C6C<M*on <&t<??.For consider the points 0, D, C) P (seefigure, p. 2) and
the projection of OP must be equal to the mm of the pro-
jectionsof OD (.=.a:),D<7(=y), and CP(=~).
8. Ha~ing estabHshedthose principlesconcerning projee-tions which we shall constantly have occasionto employ, wetetam now to thé more immediatesnbjectof this chapter.
The <!<W'<?M«t<!Mof <i5epoint f?«)~M~'Mt<%eratio m n the
<M!<<Mtce6e<tMeatwo points a:'y' <c"y"e"}are
The proof !epreciedy the same M that given at CMMM,p. 5,for the correspondingtheorem in Plane Analytic Geometry.The tmes J~ in the BgaK there given, now representthe oKtmates drawn from the two points to any one of thé
co-ordinateplanes.
6 THEFMNT.
If we consHerthe ratio m Mas mdetennîmate,we have
the eo~o-dMMtteaof <!Kypoint in the Unejoining the two given
points.
&.J[M~<t~~&M<Mt<MM:M,<M~<&e~ww~tMtj~M<~<~<~MM~w~~&e~&~ratio m+ M <0J&:J<~ <!0-0)~MOf<M~MWt of section.
~['M.
This ia proved as in Plane Analytic Geometry (see C~KMw,
p. 6). If we conmder M, n as indetermimate, we have the
co-ordinates of any point m the plane determined by the
three points.
Ex. The ImMJmmng middk pointa of opposite edgM of tettahedron
meetia~petnt. Th«'softtmMehmiddiepo:nt<Me*meet iwa point The lis of two ma middle pointe are"a a
Nidthea;ofthendddte point of the KM joining them ist?*
The other co-MdiMtMare found in t&e nxmaer, and their eymmetryahowathat thm is aho a point on thé Une joining the other middle pointe.Throu~h this came point wiN pMs thé line joining eaeh vertex to thecentre of gravity of thé oppoaite ttimgle. For thé of one of the<e
centrea of viy'+x"+x"'
centteaofgKtvityh–mdifthelinejoiah~tMetotheoppceite
vertex be eut in the ratio of3 J, we get the Mme Ttdne as before.
TKE POINT. y
11. The poaMonof a point Meome&neaexpreseed by itata&tB vector ana the angles it makes with three KcttmgotM'axée. Let theae anglesbe a, y. Thensincethe co-ordinates
a:, y, are the projections of the radias vector on the three
axes, we have
<c<=tpcoat{,y=~coa~ e–pcoBy.
And, since fe'+~K'*==~ the three cosines (which areeometunea caîî<6dthé dh-ectMn-coameaof the radius veeor)are connectedby the relation
ThepomttonefapomtisalBOBome&neaexpfesttedbythe
followmg polar co-ordmatea–thé radius vector, the angle 'y which
the radius vector makes with a Bxed axis OZ, and the angle
COJ9 ('= ~) which OC the projection of the radius vector on a
plane perpendicular to <?J?(eee figure, p. 4) makes with a fixed
line OX m that plane. S!noe then 0<7'=*p mt'y} the j~nmotta
for tKmsfbnning from rectangular to these polar co-ordinates are
<=p amycoe~, y'='~ siny mn~, e'=p cosy.
12. The <g~MM of the <XM<tof Omy~&MM~MM M e~M~ <C
~MM~W~M<~<M~~MMCmt~M<
~iMMN.
1 have Mlowed the <MM<ttpMe~ee in denoting the position of a line
by these angles, but in one point of ~iew there would be an advantege in
aang inatead the oomplementary an~tes, namely, the angles wMeh theline makea with the eo-ordinate planes. This appeara flom thé correspond-ing iiMmahe for oblique Mee whMh 1 have not thought it worth whNeto give in thé text, as we <haU not have occMian to use them tAerwardt.Let a, j3, be the angles which a line maket with the planes ~e, M, and
tet~t,J5,C'bethe angteewMchthefoit of makeswith theptaneef~,
ofy with thé plane «f < and of with the plane of then thé &tm)))mwh!chMo'espomdto thoee in the text, are
z~n~"pth<t,y~nB°~~&~ ednC"p~n~.
TheM &)nnu!e are proved by the ptincipte of Att. 7. If we pto;eet on a
lhepMptBdiMdattotheptaneof~,ahMe<hepM)eet!omtofya])d cfoonthh Une ~aaiith, the projection of < must he equal to that of the radius
vector, and thé angles madeby < and p with this Une are thé cotnptementftof Aande.
THE MMM.8
Let the area be A, and let a perpendicolar to its planemake angles a, <ywith the three axes; then (Art. 4) the
projectionsof this MM on thé planes ~e, ??, a~ teepecëvely,M'a cos~ cos~, À coa-y. And the mm of the squaresof thèsethree = eincecos*a + cos*~+ c<M~'yc=1.
18. To e~pMMthe CMMMof <~ 0~& ~e<MMM<«? ~MM<
OP, Of ~t <cntM0/'<~ <&MC<MK-OMMMNof theae~MtM.We haveproved(Art 10),
or c<M = ooMcasa' + coa~Soos~*+cos'ycoBy'.
CoB. The conditionthat two Enesehonidbe at right tmgteatoeachotheris
COS<tC08<t'+ 009~3C06~3'+COB'yC0a</==0.
14. TheMowmg&mnd&îaaIsoMmetimesuMM:
~n'~ = (co9~ coB/ cosycos/S')'+ (ooa'yco8«' cosacos-y')'
+ (coaacoBj3*coB~3coae')'.
TM: may be denTed from the followingetemeataty theorem
for the amn of the aqaarea of three (tetemNmamta(ZeMOMon
B~&er ~%fe6M!,Art. 31), but which can atso be venSed atonceby actaal exp&nsMN)
For when &, &,c; <t',& e are the ~rection-coNnea of two
Un6e,the nght-hand aide beoomes 1 eoB'C.
Et. To Cad the perpendioulardistance&oma point t-y~ to a line
)~M'eaghtheoriginwhoMduec~oa-M~eeare<~ 1.Let P be the point <y~, 00 the ~en tine, J'Q <hepstpendicahr,
thenit iaplainthat fQ°OF amPOQ; and aaingtbevaluejut obtaimdfor onPOO, andMmemberiogthat <t'=OP e<M<&o.,wehave
9T&MMMMtMATtM OF CO-0&NNATES.
l&tMctMH*cMM!M~'<tÏMMjpe)yeH<~MMt!<a'<otMW~MMB?M!M,<HM?~<M~Mjpe)peK<M[!M&N'<%<&-p&Me.
Let a' et"y" be the direetMn~ngIesof the given line,and a~Y of the required Hne)then we have to find a~S'y&<Mnthe threeequations
From the fimt two equations we can eamiydenv~ by elimi-
nating in turn cose:,cos~ cosy
This resnit may be alao obtamedas follows: take any two
pointaj~ 0, or a; a!"y"s",one on eachof thetwogiven lines.Now double the area of the projection on thé plane of a!yof tho triangle F0<?, is (Me Cbmcs,p. 26) a!y or
p'(coea' cos~cosa" cos~). But doublethe area of the
triangle te ~'p" Bm9,and therefore the projectionon the planeof xy is p'p" sin0 cos'y. Hence,as before,
sm~ cosy=cosct*cos~8"-cosa" coa~
anftin like manner
TRANSFORMATION 0F CO-ORDINATES.
16.2b<MMM/byM<0~<tMt~Moa!M<&Mt~X<t<MtC<M~Mt.to/tMeco-<M'<~M<e<f~ve(? to Medd <??&area: y, e'.
The &rmubBof transformationare (as in Plane Qeemetty)
TBAN8FOMtA'FtON<? CO-OBPÏNATE8.M
For let a line drawn through thé point P pataM ? oneofthe axes (for instancee) meet the old plane of fi!ym a point
and the new in a point C'; then FO=.J?C'+ C'<~
But jPC in the old e, PC' is the new e; and s!aoepan~Mplanes make equal intercepta on parand right !iaM, CO*muet be equal to the Mnedmm throagh the now on~n O'
parallelto the axis of e, to meet thé oldplaneofa~.
17. Tb~MtMJ~MMa ye<!<<M~«&N'systemof <M!Mto <MtO<X~
<~<<6NtO~a.CM&tWMt~MeMBM<M~M!.Let the angles made by the new axes of z, y, je with thé
old axes be a, ~9,-y; a', ~8',< a",~8", respectively. Thenif we project the new co-ordinateton one of the old axes,thesom of the three projections wHI(Art. 7) be equal to thé
projectionof the radius vector, which is the correspondmgold
co-otdma.te. Thua we get the three equations
By thé help of these relations we can ven~r that whenwe pMs&om one system of rectangular axes to another, wo
hâve, Mis geometncttîy évident,~+~ <==J!+ y*+~When the new axes are tectangaUr, amce «, K', a" are
the angles made by the old axis of <Bwith the new axes, wem<Mthave
MANMOBMATMM)0F CO-ONMWATES. 11
It would not te dMScaItto denve amdy~caHyeq~ttona
Z~&omeqBa~t~~J3,C,tmtwe6ha!lnotspendtune <mwhat ie geometHcaUyévident.
ML~V~mwoùM~MmrM~<mMto&q~~mn~
rM~h~h~X,~vt~~Ma~~Mb~wMndM!MWtm~ofy and of &anda;, ofx andy reapeo~vely,then (Art. 18)
Thnswe oMatnthera~nsvector&om&eongmtoamypointexpressedin termsof the obliqueco-OK~Dateeof that point.It is proved in like manner that the square of the distancebetweentwopomta,the axesbeing oblique,iB
19. 2%e<i~Me<<My e~M~KBe<!Mea<~ecC-<M~MM<ë<¬ a~~reaf by <nMM/M<t<MMtc/'co-o~MXœ.
Thta M proved, as at <XmM<,p. 8, from the co~detailon
thst the expresmons just given for a:, y, <, onty involve the
new co-oj'diB&tM in <&e~'a< àegree.
Aa we shall MVMrequire in pMeUee ihe fonnutte for tïMM&tntdn~from oae let of obliqueMea to another, we ontygive them in a note.
Let A, B, C h~e the Mme meaning as st note, p. ?, and !et a, /9, ~jd, ~t */< «"t ~*t be the angles made by the new aM< with the oldoo-erdinate ~~M«t then by ptojectmg on lines petpMM&mlMto the oldeo.mdmate planes, as in the note fefened te, we nnd
( 12 )
CHAPTER II.
INTERPMn'ATIONOTEQ~ATÏONS.
20. IT appesrs from t~e constructionof Art. 1 that if wewere gtvonmercly the two equations a;<=<t,y=!~ and if thee were le&indeterminate, the two given equations would de-termine the point C, and we shonid know that the point P
lay <MMM'X<Mon the Imo PC. Theae two equations thenare considerodaa repreMB~ng that right line, it being thelocus of all points whose a!< and whose y=&. We leamthen that any two equationsof the form a?=:<t,y=& representa right lineparallel to the axisof& In particdar, the equationsae=0,y==00 repreaent the axis of .s itseIR Similarly for theother axes.
Again, if we were given thé single equation a:=< wecoold determinenothing but the point D. Proceeding, as atthe end of Art. 2, we shoutd !eam that the point P lay Mme
<ote~ein the pianePBCD, but its positionin that plane would
be indeterminate. This planethen being the iocnsof all pointswhoseiB=<~!s represented analyticaUyby that equation. Weteam then that any equation of the form <B='«represents a
plane parallel to the plane y~. In pardcular, the equationo:=00 denotes thé plane yz ItseM~ Similarly, for the othertwo co-ordinateplanes.
21. In general, any singleequationbetweenthe co-ordinates
Mp~6Mt&a Mo~Meo/'wme &MtJ/any <MO<KM!<<~M!eo!M<«t(MM<~MeeM<Aeat<M'eMa<a line of MOMJM~ either a<r<t~<<M'
CM!)e~/and any <&Meequationa<&MO<!6oneM*MO~e~OM~.
I. If we are given a singleequation, we may take for a:
and y any arbitrary TahMs; and then the given equationsolved &f will determine one or more oorrespondingvaluesof a. In other worda, if we take arbitrarily any point 0 in
the plane of .cy, we can always6nd on the line PO one or
1NTEKPRETATMN OP EQUATMM. 13
morepoints whose co-ordinateswill satMy thé given equation.The assemblage then of pointa so found on the lines PO willform a aur&ee which will be thé geometricalrepresentationof the given equation(see CbKtM,p. 18).
Il. When we are given too equations,we cao, by elimi-
nating y and x altematety between them, throw them into
thé form ~==~(a:), je=~-(a:). If then wo take for a: any ar-
bitrary value, the given equations will determinecon'espondingvalues for y and In other words, we can no longer takethé point 0 <N)~!pX<Mon the plane of xy, but this point is
limitedto a certain locusrepresentedby the equation y='~ (ic).Taking the point C anywhere on thia locus, we determineas beforeon thé line PC a number of pointsP, the assemblageof which is thé locus represented by the two equations. And
since the pointa 0 which are thé projectionsof these latter
points, lie on a certain line, straight or curved, it is plain thatthe points P mut abo lie on a line of some Hnd, though of
coursethey do not necessarilylie aUin any one plane.Otherwise thus when two équations are given, we have
seen in the first part of this article that the locus of pointawhoseco-ordinatessatisiyeither equationseparately,Ma surface.
Consequently, thé locus of points whose eo-ordinates saua~'both equations is thé assemblage of points eommon to thetwo surfaces which are represented by the two equations con-mderedseparately that is to say, thé locusis the line of in-
tersectionofthese surfaces.
III. When threeequations are given, it is plain that theyare samdent to detennine absoluteiy the vatnes of the threeunhtown quantities a:, y, j:, and therefore that the givenequations represent one or more jpoM<<<.Since each equationtaken separately represents a aor&ce, it follows hence that
any three surfaceshave one or more commonpoints of inter-
section, real or imaginary.
22. Surfaces, like plane curves, are dassed according to
thé degrees of thé equations which represent them. Since
every point in the plane of «~ bas its <!=0,if in any equation
ÏNTEKMtMATMN 0F EQUATIONS.14
we make e=0, we get the relation between the fBand yco-oîttmateaof the pointa in which the plane a~ meets theBnr&cerepresentedby the equation that is to say, we getthe equationof the plane cnrve of section, and it is obviouthat the equation of this curvewill be in general of the aame
degree as the equation of the surface. It Mévident, in tact,that the degreeof theequation of the section cannot be greaterthan that of the sm&ce, bqt it appem at nHtt as if it mightbe less. For instance, the equation
M of the third degree, but when we make <=0, we get an
equationofthe eeconddegree. But since the original equationwouldhaveheenonmeaning if it were not homogeneous,everyterm mtMtbe of the third dimensionin someImeM'unit (MoCMMca,p. 61), and therefore when we make ~'==0,the re-
mMuag terma mnst atlU be regarded aa of three dimenmona.
They will form an equation of the second degree multiplied
by a constant, and denote (see .OMtt<w,p. 61) a con!c and
a line at imSnhy. If then we take into acco~ntlines at im6nity,we may say that the section of a anr&ee of the M" degreeby the plane of a~ will be <t!tM~<of the degree, andaince any plane may be made the plane of xy, and Binee
transformationof co-ordinatesdoeanot alter the degree of an
equation, we !eam that everyy&MMMc<tbtt a Mti~Meof <<eM"*d~ee is a e<<nwof the M*degree.
In like manner it ia proved that eM~ ~A< line Mee<aa
«t~ace <~tk M"*<~ee Mtn pointe. The right Une may bemade the axis of a, and the points where it meeta thé snr&ce
aïe&)tmdhymakmg<p='0,v'0!ntheeqaation ofthe sarSMe,when in général we get an equation of thé M"*degree to de-when in general we get an eqnation of tho nm degree to de.
termine < If the degree of the equation happenedto be lésathan M,it would only indioate that some of the n pointewhere the !me meets the snt&ce are at innnity.
28. C~<n)œin ~MM are daern~edaccording to the numberof points in which they are met by any plane. ~cc e~<M<<b!M
of <Jte?"' and M*~d~eM Mspecf&e~fepfeMMta ca~eeof <6e
MM**o~ee. For the sar&ces represented by the equationa
16tNTBBMETA'nON 0F EQCAtKHfS.
aM eut by any plane in carvMof the <K*"and ~"degMea
respectively, and these curvea intersectin ma pointa.Three e~<M<tMMof <M~< and 0~M yMpec<A)e~,
<ZMtO<eMMtp~OM:<This foHowefrom the theory of elimination,mmceif we
eliminatey and z betweenthe equations,ve obtain an equationof the MMF*"degree to dotefmmea! (aee ZeMMMon JS%'Ae!'
~~e~<t, p. 86). This proves ~00 that three ««~MM of (~
M* K°' <MM ye~MC<M)e~,M!<M-MC<in <!M!pjj)CM!
24. If an equation only contain two of the variables
(<c,y) =0, the letutiermight at firstsappOMth&tit representaa curve in the plane of a~, and Bothat it &tîaa an exceptionto the rule that it roquires<«?equationsto repreeent a curve.But it muet be rememberedthat the eqoation (ic,y)~0wiUbe satisSed not only for any point of thia curve in the planeof xy, but a!eofor any other pointhaving the eme <eand y
thoagh a diNeKfnt<: that ia to say, for any point of the
m~MegMMM~hyai~~tRMNM~M~thMcm~~but remaining parallel to the MMof e.* The oarve in the
plane of~canonlyberepresentedbytMM equations,namely,
e=0,~(<c,y)=0.If an equation contain only one of the variables x, we
know by the theory of equatWM,that it may be resolved
into n factors of the form a!-a=0, and therefore (Art. 20)that itrepresentaMp~esparaHeitooneof the co-ordinate
planes.
.A am-&<!9generatedbya rightUnemo~in~paralleltoUseMmca!!ett
a <yM«MM<NM&ee.
( 16)
CHAPTER III.
THEPLANE.
2N. ÏN the discasaonof equations we commenceof coursewith equationaof the first degree, and the nrst stop is to
prove that eee~yep<«<&M!e/*<%e~M<degree<~M*Men<~a plane,and conveKely,that <~ eg~M<Mwof a plane M a?M<ty<of the
~&<d5~Me. We commencewith the latter proposition,which
maybe establishedin two or three diNerentways.In the mst place we have seen (Art. 20) that the plane
of xy ia representedby an equation of the nrst degree, viz.
<=0; and tranafbmKttIonto any other axes cannot alter thé
degreeof thisequation (Art. 19).We might arrive at the same resntt by formingthe equation
of the plane determinedby threo given points, whîch we cando by eliminating ?, m, n ~om the three equations givenArt. 9, when we should arrive at an equation of thé nrat
degree. The following method however of expressing the
equationof a plane leads to <~e of the &rm8moat aseM in
pract!ce.
26. J~ ~Mf<~ eg<M~Mtof a ~&MM,MejpMyeKc~MMbfon
<pA«~J~MM origin =jp, and M<M ON~~M<t,/?) V <e&&the
oaies.The length of the projection on the perpendicular of thé
radius vector to any point of the plane is of course =p, and
(Art. 7) thia is equal to the sam of the projections on thatUneofthé three co-ordinates. Hence weobtamfor thé équationof the plane
In what MIotn we supposethe MMKetmgotM',but this equationMtntewhatevetbetheMM.
THE PLANE. 17
c
27. New, convefBdy,onyequationof the Ëf~tdegree
can be reduced to the form just given, by dMdmg it by a
&etorJ5. Weare to have~1=JBcosct,~=' JBcoe~,C==JS cos'y,whence,by Art. 11, is determiaed to be '=~'+J9'+0*).Hence any equation ~+~+C!:+~='0 nMy be identifiedwith the equation of a p!ane, the perpendicularon whichfrom
the or!gm ==//j'"B'j.'<T!) and makes angles with thethe origin
~(A, +B+ Cyand makea anglea with tho
axes whose cosmes are j4, C, respectivelydividedby the
same square root. We are to give to the squareMot the
sign which will make the perpendicularpositive,and then the
signa of the coMneswill determinewhether the angles whiehthe perpendicular makea with the positive directions of theaxes are acate or obtuse.
28. Ï~~MJ the anglebetweentwoplanes
The angle between the planes is thé eame as the anglebetweenthe perpendicnlamon them from the origin. By thelast article we have the angles theseperpendicnhmmake with
,the Mes, and theBce,Arts. 18,14,wehâve
in other words, if the coeSoients~4, B, C be proportionalto
B', C', in which case it is manifestfrom the last artMethat the directionof tho perpendicularon bothwill be the same.
29. 7b ~M< the <'g<M<Mttof a plane M <eMMof Me in-
~fcep~a, &)e, !cA<e&it makes<Mt <N!M.
THE PLANE.Ï8
The interceptBMtdoon thé axia of a:by the plane
u&MdbynMtkmgyandzboth'=0,whenwehave.~<t+.P=!0.And MmUarly,j8&+D=0, <X!+D=0. Sobstttntmg in thé
generalequationthe valuesjoat foundfor A, J~ !t becomea
If in thé general équation any term be wanting, for inotance,if ~=0, the point where the plane meets the axis of x is at
infinity,or the plane !a parallel to the axia of x. If we haveboth ~1=0, B=0, then two axes meet at infinity the givenplane which is therefore parallel to the plane of a!y (seeaiso
Art. 20). If wehave~c=0, JS=0, 0=0, all three axes meet
the phne at infinity,and we see, as at C~Mtco,p. $1, that an
equationD==0 muâtbe taken to represent a plane at m&uty.
30. To ~M <~ equation t~e ~«Ke <&<eMMM)e<?by ~~ee
pMK<t.Let the equation be ~a:+JE~+0!+2)=!0; and sinee this
ia to be satisfiedby the co-ordinatesof eachof the givenpoints,<7,D mnatsatisfythe équations
THE PLANE. 19
C2
If we constder y, e aa thé co-ordinatesof any fourth
point, wc have the condition that four pointa shonM lie in
one plane.
81. The coeSdents of x, y, e in the preceding equationarc cv!dently double the M'oaaof the projectionson thé co-ord!n&teplanes of the triangle formed by the three pomte.
Ifnow we take the equation (Art. 26}
and multiply it by twce ~t, (A being the ma of the triangleformedby thé three points) the equation wHIbecome identiedwith that of the last article, since coa<[, cos~3,JLcosyare the prqectIoM of the tnangle on the co-ordinate phmea(Art. 4). The absointe term then must be the same in bothcases. Hence the quantity
représenta double the area of the tftemgle&nned by the three
pomta moltIpUedby the p~pendiea!.ar on Ita plane from the
ong!n: or~in other worda,N&!<M)M<thevolumeo~ the <n<MtyM&ffji;yMtm«j,tc&Me&McM that ~«ï~&, and whosewy<KC& the
0!~tK.*
IfmfhepteeedingTfthtMwemtMtitttte<hr<e',y', CM«', eot/9*,CM' &e.,we6ndthet aixtimea thé winmeof thispyMmM <.dm~i, &o.,wefindthetni%timonthevolumeof dà pyramu I?prpm
multipliedbythedeterminant
Now let us Mppo<e the thtee MdU vectotex eut by a tphere whoae Kdim
h uuity, having the origin for ita centre, and meeting it m a sphetietd
triangle .B'B'JT. Then Ma denote the aide ~JB", and p the perpend~ulaï<n it from J! six times thé volume ofthe pytamid willbe ~yehKt mn~tjifor ~"<nn<t is double the axea of one &ce of the pytamid, and mn~il thé perpend!en!M' on !t from thé eppMKe ~Mtex. It MtoM then that
thé detmmtMtnt above wntten la equal to double the funetion
ofihe <MMof the &bcye-!nentionedephenealtriangle. The same thing
THE PLANE.20
We can at once express A ttse!f in termaof the co-ordmatceof thé three points by Art. 18~ and muet have 4~4*equal tothe eam of thé squares of the coefficientsof x, y, and in
the equattoa of the hst artide.
32. ?b./M the lengthof the~e~M<&'<'M&M*<MM0 given pointa: on a givenplane.
If we draw throngh iey< a plane paraM to the given
plane and let fall on the two planes a common perpendicnbu'from the origin, then thé intercept on this line will be equalto the length of the perpendicularrequired, since parallel planesmake equal intercepta on parallel ILocs. But the length ofthe perpendicularon the plane through iB'e' M,by definition,(Art. 5) the projection on that perpendicular of the radiusvector to !cy& and therefore(Art. 26) iBequal to
The tength requiredis therefore
N.B. This mppoaes the perpendicular on the plane throngh
a;y~' to be greater than j), or, in other worda, that a:y. and
may be ptOfed by forming the square of the same determinant accordingto the ot<Mnarytaie} when ifwe write
tMa" eo«t'" cM~* eo<jS" 00~ <!M' = eoBa,&c.
wMch expandedh l+2cMaeo~co&e-ec~<t-eMt't-tM*e, which isknownto havethe valueinquestion.
It ta MeM to remarkthat when the threeUaee are at f~ht anglestoeach otherthedeterminant
THËPLANE. 21
thé origin are on oppositestdeâ of the ptanc. If they wereon the same aide, the length of the perpendicalar would be
p (a)' coaa-(- y*COB~+< coa'y). If the eqoMtonof the planewere given in thé form ~.c+J~+Ck+JP, tt !a reduced tothe other form, as in Art. 27, and thé length of thé per-pendicularta
It M plain that a!l points for which ~a:'+J5~'+C~'+2)has the aame s!gn M D, will be on the same aLdeof the planeM the ongic, and cMeoer~ when the sign is dMerent.
33. ~j&t~ thec<w~M:a<esq/'<i5e<'M<M'<ec<M~:of ~~ee~&MtM.Th!s is oniy to solve three equattona of the 6Mt degree
for three NnknownqnantMes (sec ZMao~ <M1% ~i~~at,Art. 24). The value of the co-ordmateswill hecomomËnite
!fthe determinant (~(7") vanlehes,or
This then is the condition that the three planes should be
parallel to the same Une. For in such a case thé line of in-
tersection of any two would be &Tsoparallel to tMs line, and
coald not meet the third plane at any finito distance.
34. To~t:<?the co~'f<OM<Aa<four planes ahouldmeet~Ma
point.This is evidently obtained, by eliminating x, j! between
the equationsof the four planes, and is thereforethe dcterminant
(~'<7"P"'), or
36. 2~ ,/M the po~f~M<~ MM~fM M~OM<w<M;Mare
<tNyJ~M~ytCeMJMMt~.If we multiply the area of the triangle formed by three
points, by the perpondicular on their plane from the fourth,we obtain three tunes the volume. The length of thé pcr-
22 TKË PLANE.
pendicahu'oa thé plane whoae equation !s given, (Ait 80) M
formedby «abstttntmgin that equationthe co-ordmttteaof the
ibarth point, and dividing by thé sqttaMroot cf the aorn of
the squaresof the coûSdente of x, y, e. But (Art. 81) that
square root M double the area of the irtangle formed by the
three pointe. Hence s&! timea <~ volumeof the t~a&a&WK
in gMM<MMM<~<Mt7? the <&<enHMKMt<
86. It is évidenteu in Plane Geometry,(aeeC~Mt&w,Art. 36)that if ~S", represent any three surfaces, then a~+M"wheMa and b are any constants, represents a aor&ce pasamg'through the Une of intersecëon of B' and and that
o~+M'+cN" représenta a surfacepaastag throngh the pointsof intersectionof S, jS',and <8' Thus then if Z, N denote
any threo planes, aZ+~~f dénotes a plane passing throughthe line of intersection of the amt two, and <!tZ+~Jtf+<denotes a plane pammg through the point common to all
three. As a particular case of the preceding oZ+6b denotesa planeparaBelto Z, and <~ + Mf+c dénotes a plane parallelto the intersectionof L andJhr (seeArt 29).
So again, four planes Z, J~ P will pass throngh thesame point if their equations are connected by an identicalre!ation
Thevolumeofthe tetmhedronformedbyfourplanes,whoseeqm~MMara given,m be foundbyformingthe co-otdmatesof its angular pemte,and then mtMtttntin~in the formula givenabore. The TMnttie, (teeJ~M<MMonJB%«' ~M, Art, 26) that <!xtimes thé volumele eqcal to
where B h the detenamant (~JB'C"JO~Art. 34, and the &et<tnin thedenominatorexp[M<the ecndMoM (Art. M) that any three of théphne<ahouldbe pM-attetto the aMieline.
THE P~ANE. 23
for then any co-ordtMteswhieh satidy the &*st&Ke mast
aatlsiy thé fourth. Converaety,given any 6m)'planes mter-
Bectmgin a commonpoint, it Meaayto obtMnmch an identicalrelation. For multiply thé &rst equation by thé determinant
(~C"'), the eecondby- (~B'"C), the third by (~j8C"),and thé fourth by ~'C"), Nidadd then(JEeMOM<M~%rXe!'Algebra, Art. 7) the coefficientsof if, y, e vanieh ïdenticaHy;iaad the remaining term M the determinantwhich vamahes
(Art. 84),because the planes meet in a point. Their équationsare thereforeconnectedby the identicalrelation
87. Œven any four planes Z, JM, P not meeting in a
point, it ia easy to see (as at <XMtM,Arts. 68, M) that the
equation of any other plane can be throwninto the form
And in general the equation of any sm&oeof the M*"degreecan be expresaed-bya homogeneousequationof the M"'degreebetween Z, J~,P (see OMt~, Art. 270). For the numberof terms in the compoteequationof the order between tt~eevariablesM)the same as the namberof termain the A<M)M~e<MO!Mequation of the M'~orderbetweenfour variables.
Accordmgly, in what &)tlows,we shall UM these g<t<M?!
planar co-ordmateawhenever by so doing OttreqMlioBBcan
be materially simplified.
Ex. 1. Tofind,.theequationof the ptane.pMaingthroughz'e', andthroughtheintersectionofthephnea
Ex. 3. Fiud the equation of the plane pMmngthrough the points~l~C, figure, p. 2.
The equatioM of the line BC are evidently '=-1, ?+=1. Hence
obviouoly the equation of the required plane !<<t
+6 + e
2, Bince this
pMMa through each of the three linesjoining the three given points.
24 THE KtOHT UNE.
Ex.9. Kndtheequationofthéplane~'J?yinthéMunefigure.
TheeqaatioMofthélineBF are~O~+'~t, andformingMabove
the equationof the planejoMng tbiaUneto the point été, we get1!+ x 1.~i.e e «
38.MM*plana which M<M'~C<in a: right line be metbyany plane, f~eoMAafmottMM!<Mof thejMaot~<oj~MMed'will &6constant. For wecouHby trans~nnfttton of co-ordinatesmake
the tMuosveKeplane the plane of icy,and then by making ~=00in the eqa&tMHMwouldhave the equationaof the intersectionsof the four planes with this plane. Thèse will bc of the form
<tZ+J~ M+J~ cZ+3~ d'C+Jt~ whose anh&nnon!oratio
(aeeC~Ktca,Art. 56) depends soletyon the conatants< b,c, d;amddoes not alter when by tranaformattonof co-ordinatesL
and Jf come to represent dt&a'ent lines.
THE RIGHT LINE.
89. The equaûona of any two planes taken together will
represent their Une of intersection which will include all thé
pointswhoseco-ordinatessatis~ both the equations. By el!nn-
nating x and y alternately between the equations we redacethem to a form commontyused, YM.
The firat represents thé projection of the lino on the plane ofa:~and the secondthat on the planeof y. The roader will ob-
serve that the ~a<MtMof a right line McMs four <MepeM<&!M<co~aH<S.
We might form independently thé equations of the line
joining two points; for taking the values given (Art. 8) of the
co-ordinatesof any point on that line, solving for the ratio
wt ? from each of the three equations there given, and eqna-
ting rMaIts,weget
for thé requiredéquations of the line. It thua appears that
TH)S RMHT LINE. 25
the equations of the projectionsof thé line are the sameMthe
equations of the lines joining the projeotionaof two points onthe line, M is otherwiseevident.
40. Two right lines in space will in general not intersect.If the Ont line be represented by any two equationsL!=0,
if'~0, and the secondby any other two ~=0, J?==0,then ifthe two tmea meet in a point, each of theae four phnes mnst
passthrough that point, aad the conditionthat thé Unesahouldintersect is thé same as that already given (Art. 84).
Two imtersectmg Unes detemune a plane whose equationcan eas!ly be found. For we have seen (Art. 86) that whenthé four planes mtemect, their equations satisfy an identicalrelation
The equations therefore <tL+&3f=0, and c~+<!F=0 mustbe identical and muat represent the same plane. But the formof the first equation ahowsthat this plane pMseathrough the
UneZ, and that of the secondequationehowsthat it paBseathrough the line JR
Ex. WhenthegivenlinesMerepreaentedbyequationsoftheform
the condition that they ahould intersect is easily found. For solving for a
from the &st and third equations, and equating it to the value found by
solving &om the second and fburth, we get
a-d t-~
m-m' ?-?'
Again,if this conditionin aatisfted,the fourequationsmeconnectedbythe identicalrelation
h the équationcf the planeeontamhtgbothUnes.
41. 2~jM the egaa~M<Q~MM~d~WM~Athe ~o&t<
a!y< cand MMtJ!K~angles ct~, 'y Mt'<~ .a~~The projections on the Mes, ~BMttM of icye' from
any varlaMo point N~~Ene,arc reaMcttvety a?-a!
y- e-z'; aud sinco thes~Mc cfM~e(M~rto that distanco
26 THE BMHT UNE.
multipliedby thé cosïneof the angle betweenthe I!noand thé
axis in question,wehave
a form of writing the equations of the line whicb, atthough it
indudett two anperSaoaaconstants, yet on aceount of ita eym-metry between a~y, o ï8 often tMedin pM&renceto the fbnninArt.89.
RectprocaUy,if we deaire to Endthe angles made with the
axes by any line, we have only to throwtta equation into the
0-d w-té a-a'form
–T– == when the diiecdon-cosmesof the
~L jLt (~
line will be respectively B, C, each divided by the squareroot ofthe sumof thesqtmtesof thesethree quantities.
Ex.1. ToSndtbedhreetion-cosiBMofo!"M«to, yc<Mt&. WtMtig
the equttttOMin the fNam *-L?~o~ thé directton~:OMae<areM ft t
ElimhMtm~y and t <dtem&telywe redocethemto the precedingform,jBC'-J3'C <M'-C'~ .~y-~B
Wb«61
<tndMeQMee<ton-eeMnexn'e ––s– ––s––< where~t ~C ~t
ia thesumof theequarettof the threenametatoN.
Ex. 4. To Bndtheequationof the ptanethtoaghthe twointeKeetmj;lines 1 1 »
The required phme pMSMthrough ~yy and !ta perpendieuler la pMpen*dioular to two lines whosedirection-aosines are given} t&Me&re,(Ath 16)the required equation is
(«! .)~)(CM/9CM'/ CM~ COf~) + (y /) (<M'yCM«' cosy cosa)
THE BMHT MNE. 27
tt)b~Ta&!dtheeqaatiMofthetJMMt'Mdag&Maght!MtwopMaMUnes
The teqnited plane contaiM the line joining the given pMntt, whosedireetien~esinM Me proportional to o:' < y' «'-< thé diteetîon.eodnM of the perpemMeuItu*to thé plane Me thoMt~MpMport!on<ttto
TheM may therefore be taken M the eoeCeienh of ar,y, <, in the tequitedéquation, wMte the ttbMintB tena detMm!ne<Iby Mbâdtati!~ jey~ for
intheeqcttîonie
42. ïbj~aJ<Xe e~~M<MMO~~jpetyeM<Kct~~OMa:y<on <Xe~M .~a'+I~+Ok+.D. The direction-eosines ofthe
perpendicular on the plane (Art. 27) are proportional to ~i, B, 0;hence the equations required are
43. To~K<?~Ae~M'ectfM-CMMetof the &Meo<M'of the <Mt~e
&<!<M~(<W~K)ett?MMN.As we are only ccncemed with d'M'«~o)Mit is of course
safMent to conaider lines through the origin. ]? wa take
pomts a:y< a:"y"< one on each line, equidistant from the
origin, then the middle point of the line johing these pomtaM evidently a point on the Msector, whose equation would
therefore be
and whose~rect!on-«Mtneaare thetetENraproportionalto
bttt NNooa; y', <B", are evidently pMporhonalto thetHrection-comneaof the given lines, the direction-comnesof theMacetorare
eosa'+cosa", co9<8'+cos~3",coa'/+co8'
Mch divided by the square root of tho aum of the sqmu'eaof
theae thrce quauttttes.
THE RIONTUNE.M28
The Msector of thé supplemental angle between thé !met
woald bc got by substituting for thé point a:"y"< a point equ!-distant from the origin measured in thé opposite dîrectMnjwhoseco-ordinatesare .B", -y", and thereforethe direc-
tion-cosinesof this bisector are respectivelyproportional to
N.B. The equation of the plane bisecting the angle batween
two given planes is found precisely as at CMtt~M,p. 85, and M
(a!C08a+ycoa~+~co8'y–~)==i(a:cosot'+yco8~'+~co8y').
44. 2b find theangle made with &M&other by «eo linea
Evidently (Arts. 13,4t),
CoB. The lines are at right angles to each other if
y-t-MMK'+KH'=0.
Ex. Toflndtheanglebetweenthe ânes 1/ 0 98-0.Ex. Toandtheanglebetweenthe
Une!! ––r .) ––r==y, =0.
Ana. 3~.
4S. To ~t<~ the angle between the plane Ax + Ck + D,a!o y"& <c
<!M<~ &Me =' =< Mt <!
The angle between the line <mdthe plane is the complementof the angle between the line and the perpendicular on the
plane, and wo have therefore
CûB. When ~+&M+C~=0, thé Une is parallel to the
p!ane, for it !a then perpendîcnlar to a perpendicular on the
plane.
46. To find <&<CMM&MoKathat a line a!==M!J!+a, y=!~M+tb
ehould &eot~~er M? ~&Mf' ~.x+J~+Ck+jO. SabatHate
TanE RMHT LINE. 29
for<eand y in the equ&ttonof tho plane, and solve for <, whenwe have
aad if bothnumerator and denominatorvanish, the value of eis mdetermmateand the line Maltogether in the plane. We
have j<Ntseen t!Mt the vanishingof the denominatorexpresseathe conditionthat the line shouldbo pM~el to the plane; wbilethe vaniabingof the numeratorexpremesthat one of the pointsof the lineis in the plane, v!z.the point <t6wherethe Imemoetethe planeof a?y.
In like manner in order to nnd the conditionsthat a rightline ahouldlie altogether in any surface, wc abouldmbstitntefor a: andm the equation of tbe sur&ce, and then equate tozéro the coefficientof <e~ powerof <!inthe remMng equation.It is plainthat the number of conditionsthue resulting is one
more than the degree of the onr&ce.*
47. ToJ&«~theequationof the~&MMdratm througha ~M)Mt
KMejpe)yeK<~MM&M'toa ~~M~?<!)M.Let thé line be given by the equations
Since the equatioM of a right Une contain foor constante,a ïi~ht linecan be detetnuned whieh eMt MtM~ My &a)' eoNd!t!cna. Hence anyMt&ee of the second degree mmt contain an Mntty of right lines, aincewe hâve only three eondidom to MiMy and bave four constants at ouf
di<poM]. Every Mï&ee of the tt)M degree m<Mtcontain a nnite number
of right lines sinee the number of conditions to be MtM&edis equal to the
number of duposaMe constante. A surface of hlgher degtee nri!t not
neeeMmBycontain any right lime lyhtg altogether in the autËMe.
THERMBTUNE.30
This equationdetemunesX Mtdthe equationof the requiredptaneM
we can otherwiseeasily détermine the equation of thé requiredplane. For it is to containthé givenline whosedirection-anglesare ft', y; and it MaIso to contain a perpendt"ahu*to the
given planewhosedirection-anglesare a, ~3,'y. Hence (Art. 15)the d!rect!on-coBmesof a perpendicularto the required plane are
proportionalto
COS~COSy-COS~COS~COSy'COM-COS~COM',COMt'C0~8–C08C[CO)~8',
and since the required plane is a!aoto pasa throïtgh a:y<} ita
equationîe
48. <<!M?MM<<O~K~<<5e~!M<M!t~'a~&M)e<&~MOK
<j5MK~&e«&ef~<tMt!M<c <~ o<
First, let the given lines be the intersectionsof the planeaZ, M; J~ P whose équations are given in the most generalform. Then pïoceedingexactty as in Art. 36) we obtain the
identical relation
the right-bandmdeof theequation beingthe determinant,whose
vamaMage~reœes that the four phmeameet in a point. It is
evident then that the equations
representparallelplanesmncetheyonJydiSM'byacomt&ntq)Mndty;hïttheaeptaajespMsettchthK)agh«neof~eg!venlines.
TM RKHtT M!fE. ai
Sec<MMt!y,let the Unesbe given by eq<mii<m8ofthe form
Then mncoa perpendicular to the sought plane iaperpendicularto the directionof each of the given lines, its du'ectMtMOBmea
(Art. 15)are thé same as those given in the last example,andthe equationsof the soughtparatlel planesare
Thé pefpendicaJardistancebetween twoparallelplanesis equalto the differencobetween the perpendicularalet &!1on them6'omthe ongia, and is thereforeequal to thé dISerencebetweentheir absoluteterme, divided hy the square Motof the snmof
thesqaareBofthecommoncoe&cientsofa' Thuatheper-
pendiculardistancebetweenthe planealast foundis
where 6 (seeArt. 14)is thé angle betweenthé directionsof the
given l!nea. It is evident that the perpendiculardistancehemfound is ehorter than any other line which can be drawn 6'om
any point ofthé one plane to any point ofthe other.
49. ïb ~M <~ eg<M<t<MMand Me NM~M«M&of the <i6o!~a<<&<&MMete~tCeM<«?KCM-MttM'MC~Mt~~Mt.
The ehortest dMtancebetween two Imeais a !mo perpen-dtcalar to both, and which can be found a follows: Draw
through each of the lineB,by Art. 47, a plane perpendicularto either of the parallel planes determinedhy Art. 48; then theintersection of the two planes ao drawn will be perpendieolarto the parallel planes, and therefore to thé given Uneswhtchlie in these planes. From the constntctionit is evident that
NOTE ON THE PBOPEBT!E80F TETBAEENtA.M82
the line so determined meets both thé given lines. Ita mag-nitudeM plainly that determinedin the last article. Working
by Art. 47 the equationof a planepammgthrough a linewhese
direction-anglesare <t,~8,'y, and perpendicularto a planewhose
direction-cosinesare proportionalto
cM~coa~-tos~coey)co~y'costt-coaycoaa',coact'co~–cosacoa~
we findthat the linesought ts the intersectionof the two planoa
The direction-coMnMof the ahortest distancemuât plainty be
proportionalto
coe~cos~-coa~coBy,cos'y'coatt-cosycosa',cosft'cos~-eûsacot~S'.
NOTE ON THE PROPERTIESO? TETBAHEDRA.
60. We add aaan appemdixto the preceding chaptemsome
propertiea of tetrahedra which, though not obtained by the
methodof co-ordinates,are worth being set down.To~)d' the M?<t<MM5«<MeMtheNM:MMMJO~ttKyany four
points in a plane.Let et,b, e be the aides of the triangle fonnedby anyI~N'ea
of them ~jB<~and let <~e, f be the Unes joining thé fourth
point D to thèse three. Let the angles anbtended at D bya, b, c be a)~ then we have coset!=oos(~3±'y),whence
coa?'<t+cos*t cot~'y-3 cosacos~ co9')"=l.
This reiation will be troe whatever be the podtion of D,eitherwithinor withoutthe tnangle ~-BC'. But
NOTEON THE PROPERTÏESOF TBTBAHNHtA. 88
u
Snbsdtntmgthèse values and redudng, we findfor thé requiredMhttton
61. Tb ea~pMMt~e eo~tme of a
M'.cedges.Let the aides of the triangle formedby any face ABC ht
<&,< the perpendtcalar on that face 6'om the remainingvertex be and the distancesof the foot of that petpendicuiMfrom~1,B, 0 be d', e',f'. Then a, b, c,d', e', are connected
by the relationgiven in the last article. But if <~e,y be the
remaming edges <y='<?"+~ e*='e'*+~ y ='+?*; whenct
~e*=<e", &c.<mdpntting in thesevalues,weget
where .F Is the quantity on the teA-handBideof thé equationin the last artide. Nowthe quantity multiplying !a16 timesthe square of the area of the triangle ~LBC,and emeep mul-
tiplied by this area ia three timeathe volumeof the pyramid,wehaveF=-MF'.
52. ïb ~tJ the ye&t<MM&e<<MeHMe OM!o)~ joMM)~fourpointeomthe <tM~iteeof a sphere.
We proceedptedaely as in Art. M, only sabetitatmg forthe6)rmuhethere used the coMespondiagfbnnuto for aphericaltriangles, and if fc,~8,y, S,a, represent thé coMMetof the sixaNs in question,weget
NOTE ON THE PROPMTïEa 0F TETNAHBNtA.84
Now from this matrix we eanform (by the method of ZetMM
o» JBiyJtef~~e&Mt,Art. 20) a determinant wMchahall vanish
Ment!caUy,and wMch (aabstitatmg coe'Ct+cos'KMB'yl,cosacoaa'+ cos~coB~3'+ cos'yco8'/ = coaa&,&c.)H
which expandedbas the va!uc wntten above.
63. To~M~the MtdSMMof <~ <pAe~C<fC<MM<!rtMKy« <e<Mt-AedSroN.
Smce any sMea of the tetrahedron !a thé chord of the are
whoBecoMneisa,wehttvea=l-
with Mm!!arexpMsmoDa
&t 'y,&c.; and makmg &eM enbstitu~ons,the tommia of
the last examplebecomes
The reader may exerciseMmselfm proving that the shortest.dietMtCûbetweentwo oppositeaidesof thé tetrahedron is equalto suc times the volumedividedby the product of those sides
m~Mpliedby the sineof their angle of inclinationto each other,whichmay be expressedin tenns of the aidesby thé help of thérelation2ad coaC=' &'+< °.
( 3S)
D2
CHAPTERIV.
<PROPERTBES COMMON TO ALL SURFACES OP THESECOND DEO&EB.
64. WE shall wnte the general equation cf the second
degtee
This equation contains t~mtenna, and sinee Ita significationienot altered if by divisionwemake one of the coefBeîentsunity,it appears that nine conditions are sufficientto determine asurface of the second degree, or M we ahaUcati it for short-
neaft,a quadricsurface. Thus if we were givennme points onthe aur&ce)by sabstitattng saccess!velythe co~ordmateaof each
in the general equation, we obtain nine equations which are
snNident to determine the nme unknown quantities ") &c.a' M
And in like manner thé number of conditionsnecessary to de-termine a sorface of the ?" degree Mone leœ than the numberof terme in thé general equation.
The equation of a quadrio may abo (see Art. 87) be ex-
pressed as a homogeneonsfonction of the equations of four
given planes a:,y, z, «t,
For the nine independentconstante in the equationlast written
may be so determined that the surfaceshall paMthromghnme
givenpoints,and ihere&Nmay coincidewithany given quadric.la like manner (eee C~Mtc~p. 68) any ordmary x, &équa-tions may be made homogeneousby the introdaction of the
Théreaderwillcomparethe aorrespondingdiMa<a!onofthe equationof thé seconddegtee(C~«M,p. H9) and obMtwthé identityof themethodanowpasued andof mamyof the reMl<aobtained.
86 MM'ERTIESCOMMONTOALL8CKFACES
linear unit (wMchwe ehaB call w); and we ahall &eqMnt!yemploy equationswntten in this form for thé saké of greater
symmetry in the results. We shall however for simplicitycommencewith .< y,< co-ordiaateB.
M. The co-ordmateaare trMM6)nnedto any parallel axes
drawn tbrough a point a~'e', by writing a!-t-iB',y+y*, <;+~'for x, y, respectively(Art. 16). The result of this eubsttta-tion will be that the coefficientsof the highest powersof thevariables (a, 6, c, ?, m,N) w!U remainunaltered, that thé new
absoluteterm willbe F' (where P* ia the reault of eubetituting
a)',y*,e' for a;,y, e mthe given equation),that thenewcoeS-
66. We can transform thé general equation to polar co-ordm&tesby writing!C==~p,y'='< ~==<%(where,if thé axes
be rectangular,~4,J3,C are eqnal to eosa, <OBj8,cosy respec-tively, and if they are oblique (seenote, p. 7) ~4,J9,<?are Bt!U
quantities dependingoniy on the angles the lino makes with
the axe:) when thé equation becomes
This being a qoadr&ticgives two valuesfor the length of theradius vectorcorrespondingto any given direction; and since
any point may be taken for origin it proves that everyright~MMMee<ta g<«!<Mcin two jMMt<t,as was proved alMady(Art. 22).
67. Let us consideriSmtthe oaMwhere the origin iaon the
sttr&ce(and thereforeJ=0), in whiehcase one of the roots ofthe abovequadratieis pc-0; and let us seek the conditionthatthe radius vector should tonch the Mrfaco at thé origin. Inthis case obviouslythe second root of the quadratie will aiso
vanish, andthe requiredconditionis there&re~ +~+y(7=0.
0F THE SECONDDEOME. 87
and evidently expressesthat the radius vector lies in a certain
fixed plane. Andsince ~t, B, C<uresnbject to no restriction
but that already written, everyradins vector throughthe origindrawn in this plane toacheathé sur&ce.
Hence we Ie<umthat at a given point on a quadric an in-
Ëntty of tangent line8can be drawn, that thèse lie all in one
plane whichis caUedthe tangentplane at that pomt; and thatif the equationof the surfacebe written in the&rmM,+M,e=0,then «, ci0 is the equationof the tangent planeat the origin.
68. We can findby transformationof co-ordinatesthe équa-tion of the tangent plane at any point .ey~ on the surface.For whenwe transformto this point as origin the absoluteterm
vanishes,and the equationof thetangent plane ia (Art. 06)
TMamay he written in a more symmetncatform by tho intro-dnctionof the linearunit m,when,mmceit !anowa homogeneonsûmodon,anAsincea:y< Mto aatta~ the equationof the sar&ce,wehave
MOPENTIBaCOMMONTOALLSURFACES88
Thisequation,it will be observed, Meymmetnod betweeni~and a~y*<and maylikewisobe written
B9. ~M the point of coM<ac<of a tangent ?« <wjp&t<M<&a<Mttib'ct~Aa ~tMKpoint iB' tM<OK<&e«M~tce.
The equationlast ~mid expressesa relation betweenthe co-ordm&tesof any point on the tangent plane,and a!'y'e'M'its point of contact; and tance now we wiahto indicate that theformer oo-otdinatesare given and the latter sought, we have
only to removethe accents from the former and accentaatethelatter co-ordinates,whenwe find that the pointof contact mustlie in the plane
which ia called the pdar j!!<MMof the given point. Sincathe
pointof contactneedsatis~r no other condition,if we take <Htyof the pointswhere the polar plane meets thé surface, the tan-
gent plane at that point will paœ throagh the given point; andthé Unejoining the point of contact to thégiven point will bea t<mgeatline to the MtËtce. If ail the points of intersectionof the polar plane and the surface be joined to thé givenpoint, we sha!l have all thé lines which can be drawn thronghthat point to touch thé surface, aad the assemblage of thoselines form what is called the tangent «MMthrough the givenpoint.
N.B. In general a aurface generated by right lines whichall pass through the same point îs caUeda cone,and the pointthrough which the lines pass ia called its ee)'&a;. A cylinder(aeep. 15) is the limiting case of a cone when thé vertex is
innnitelydistant.
60. The polar plane may be abo definedas the locns ofhannonicmeans of radii passmg through the pole. In &ct letus examine the locus of points of harmonic section of radii
passingthroagh the origin then if p" be the roots of the
0F TBESECONDNMMNE. 89
quadratiucf Art. S6, and p the radius vector of thé bcM, weare to have
but this is exactiy the polar plane of the origin, M may bosoen
by making a; y', a' ail !=0in thé equation written in full
(Art. M).From this definitionof the polar plane, it is evident that if
a sectionof a surfacebe madeby a planepassing through anypoint, the polar of that point with regard to the section willbe thé intersectionof the plane of sectionwith the polar planeof thé gtvoa point. For the locus of harmonie means of «HKntiipassing throngh the point, must include the locus of har-moniemeanaof the radii whichlie in the planeof section.
61. If the polar plane of any point ~t pass through jB, thenthe polar plane of B will paMthrongh
For since the equation of the polar plane is symmetncalwith respect to a!y<ai'y~ we get the aamereault whether weeubstitatethe co-ordinateaof the eecondpoint in the equationof the polar plane ofthe first, or wice<w<
The intersection of the polar planes of Aand of willbe
a Unewhich we ahall call the polar line, with respect to the
surface, of the line ji&It ia easy to aoe that the polar Uneof the line AB ia the
locus of the poles of all planes which can be drawn throughthe line ~&
62. If in the original equationwe had not only J='0, but
atso jp, q, r each =0, ~en the equation of the tangent planefound (Art. 68)becomeaillusory,mnceevery term vanishesandno angle plane can be called the tangent plane at thé engin.In &ct the coefficientof p (Art. M) vanisheawhatever be thedirectionof /), and thereforeeueryline drawntbrough the originmeets the surface in two consécutivepoints, and thé origin isMndto be a doublepoint onthe surface.
40 PMPEM'tESCOMMUNTOALt SURFACES
In thé present caae, the equation dénotée a cone whosevertex!a the or!g!n,as in &M!tdoesevery homogeneousequationin a:, e. For if such an equation be Mtis&edby any co-
ordinateaa~,y', e', it will aiso be MtMËedby the co-ordm&tea
JB.)!')J!y, (where-Bts any oonetamt),that M to My,by theco-otdm&tesof every point on the Ime joining a''y'e' to the
origin. This line then lies wholly in the surface whiehmustthere&Mconaistof a ttatMSûf right linea drawn tbrough tha
o!<gia.Thé equation of thé tangent plane at any point of the
cône nowunder considerationmay be written in either of the
forma
Thé former form (wanting an abaolntotenn) ahews that the
tangent plane at every point on thé cônepaaseathrongh the
ongin;thel&ttet&nn6hewBth&tthet<mgentpIsne&t&nypoint iey~' touchesthe mrface at every pointof the linejoiningaiye' to the vertex; for the equation will represent the 6ame
plane ifwe eabstitute~c', ~y', Ba' for a! y',When the point a;ye' is not on the surface,the equationwe
havebeemlast diBcaemngrepresents the polar of that point,andit appears in like manner that the polar plane of every pointpaasesthroughthe vertex of the cone, and a!so that aNpointawhichlie onthe Mmeline passingthrough the vertex of a conehave the samepolar plane.
To find the polar plane of any point with regard toa conewe need only take any section throagh that point, and takethe polar limeof the point with regard to that section; thenthe plane joiningthis polar line to the vertex will be thé polarplane required. For it was proved (Art. 60) that the polarplanemustcontainthe polar line, and it ia nowproved that the
polar planemuetcontainthe vertex.
63. We can eaailyfind the condition that the general eqna-tion of thé seconddegree ahouldrepreaent a cône. For if it
does it will be poamMeby transformationof co-ordinatesto
OF THE SECONDDËOBEE. 41
make the newp, f, <~vamah. The cû-ordm~teaof the newveftex must therefore (Art M) aatif~r the conditions
which,written at jMltength, M
whichis the J&cWmMMHttof thé given equation (see ZeMMMon
~~H, p. 44).
64. Let na return now to the quadraticof Art. S6,in whichd Mnot supposedto vamah, and let us examine the condition
that thé radius vector should be b!aectedat thé origin. It !s
obviously necesaaïy and mtiBctentthat the coeffident of p in
that qaadratM should vamah, since we should then get for pvalues equal with opposite signa. The condition requiredthon is
whieh multiplied by p ehewsthat the ra~ua vector must lie m
the plane ~.c+~+~~O. Hence (Art. 60) every liM
JM!!CK<&f<M<y&the O~Mt Mt<tplane jM~tNe!to t<<~0&M'plaMM&Mec<N~a<theorigin.
66. If however we had p'='0, g=0, f==0, then ec<ry tme
drawn through the origin would bc btsected and the origin
MOPENTIES COMMONTO AU. SURFACES42
wonMbe<!tJIe(tthe<xM~oftheftat&eû. JS~tKMfr&~M~t generalone (M)<!but <MMcentre. For if we eeek by ttMM-
formationof co~ïdtnateato makethé newp, q, ~'=0, we obtainthreeequations,'n&
If however S'~00 the co-ordm&tes of the centre become infinite
and the surface bas no finite centre. If we write the original
equation M,+<t~+«.'=<0, it !a evident that8 is the disariminant
ofM,
It !a poMtNethat the nmmemtmNof theae &<MtioMmight vaniah atthe eme time with the denominator. in whicheaae the c<x)tdimttMof thecentre would become indeterminate, and thé aut&ea woald have an intaity
U du Uof Mntres. Thus if the three planes ~t
all pan th~ough théd.v p
same line, any point en this line wiNbe a contre. The eenditiona thatthb should be the caee may be written
the notationindicatif that all the four determinantamust"0, whiobare
got by erasing<myof the vertical MNM.We shall reuerve the MIm die-cuMtoaof theseeaM<forthe next chapter.
OF THE SECONDDEONEE. 48
66. ~NM~C~~JM~M
toa givmœ CCP= a
«~tceM~e <= = C,.If we trams&trmthe equation to any point on the locm M
origin, the new r must &MI the condition (Art. 64)~+gB+)'C==0, ond therefore (Art. 6&)thé equation of thelocus is
This denotes a plane through thé intersectionof the planesdU dU dU
-.r-, -y-,that la to eay,through the centre of the surface.
It is called the diametral planeconjugate te thé given directionof the chords.
If a!y<' be any point on the radins vector drawn throughthe origin parallel to the given direction, the equation of thé
diametral plane may be written
divide itbyJB,andthenmakeJBinnmte,ve8ee that thédiametral plane is thé polar of thé point at Innnity on a Unedrawn in the given direction,M we might also have inferMdfrom geometrical oonatdeMtttom(aee C~Mw,p. 272). In like
manner, the centre ia thé pole of thé plane at infinity, for ifthe origin be the centre tta potar plane (Art. 60) is d=0,which (Art. 29) repreaentBa plane mtttated at an înnmtediatance.
In the case vhere the given surface iBa cône, it is evidentthat the plane which bisectschordeparaiM to any !iae drawn
through the vertex ie the same as the polar plane of anypomt in that line. ~i &ct it was proved that ail points onthe l!ne have the same polar plane, therefore the polar of the
PMPEBTIES COMMON TO ALL 8PBFACM44
point at m6n!<yon that line is the eame M the polar pîamoof any other point in it.
67. The plane which bisects ohoKtaparallel to thé axis
ofa: Mfoundby making ~'=0, C!=0m the equation ofAtt 66,to be
and this will be parallel to the axis of y, if K=0. But this
Mabo the conditionthat the plane conjugate to the axis of yahouldbe parallel to the Mas of Hence the plane coK-
jugate to a ~tMMdirection beparallel to a <eo<M~given ?«,thej)&MMo(M/M$M<eto the latter <c<?~&eparallel to<Xej~MM*.
When ~=0 the axes of ;c and y are evidently parallel toa pair of conjugatediameters of the sectionby the plane of ay;and it is otherwMeévident that the plane conjugate to each
of two conjugate diametera of a section passes through theother. For the locaa of middle points of <tKchorde of the
surfaceparallel to a given !ine muet inelude the locae of themiddlepoints ofaU snch ehordswhich are contained in a given
phne.Three diametral planes are oaid to he conjugate whm each
is conjugato to the intersection of the other two, and threediametersare sald to be conjugate when each is conjugatetothe plane of thé other two. Thus we should obtain a systemof three conjugatediameters by taking two conjugate diametersof any central section together with thé diameter conjugateto thé plane of that section. If we had in the equation~==0,Mt!=0,n=0, it appears &om the commencementof this articlethat the co-ordinate planes are parallel to three conjugatediamétral planes.
It &Uow<that the plane.); 0 wiUbtMCtchordeparallelto theMthof if <t 0, m 0, 0) or, in otherwordt,if the originalequationdo notcontainanyoddpowerof But it le otherwiseévidentthatthMmastbe the casein order that for anyasaignedvaluesof y and<wemayobtaincqualand oppositevaluesof .)'.
0F THE 8ECOSD DEGBBE. 46
When the surface is a cone tt ia evident from what was
said (Arts. 62, 66) that a ayetemof three conjugate diametersmeets any plane section in points such that each M the polewith respect to the sectionof the linejoiningthe other two.
68. A diametral plane ia said to be principalif it bo per-pondtcular to thé chordeto which it ift conjugate.
The axes being rectangular, and 2?, C the direction-comnesof a chord,wehave seen (Art. 66)that thécorrespondingdiametral plane is
and this will be perpendicnlar to the chord, if (Art. 42) thé
coefficientsof a;, y, s be respecttTelyproportionalto ~i, B, C.Thisgives us the three equations
J[a+&!+C~=~ ~n+~+C?=AB, ~M+N+Ck~JSC.
From these eqaatîons which are linear in jB, C, we caneliminate~1,B, û, whenweobtain thé déterminant
And the three values henoe found for JB being snccesBtvetysubstituted in thé preceding equationsenable ua to determinethe correspondingvalues of j4, J9, C. Rence a gMa~nc~t ~etMMJ~<~ejM'tMCtp<t~d'M!a!<<r<t!jp&MM,the three diameters
perpendicularto whichare called the <N:Mof the surface. We0shalldiscuesthia equationmorefally in thé next chapter.
MtOPNfnES OMWON TO AM. SURFACES46
If!ntheMweMMteta<R"8,weaBd~=~-CL MuMptyingty~,and mtxHtath~<*ht &e.,weget for thé equationûfoneof theend aubetitutingmfor .dp, &o.,weget for the equationsofoneof theMMi~yo. & AadthephmeQKwnthmughtheo)'igin,(whi<!hhtheeentfe)pMpendieatMto &JeMne,la <+2y+&t ?. In likemannertheothertwoprincipalplanesare2j): 2y+<c 0, 3~ y &) 0.*
69. The MC<M!Mof a S~M<M!~yp<M'a!Mjp!<M!M«M~t&K'
<c«MXo<tef.Since any plane may be taken for the plane of a:y, it M
anScîent to consider the sectionmade by tt, which is found
by putting <!=0 mthe equationof the sm'&ce. But the section
by any pM&Uelplane is found by trtma&rmiDgthe equationto paraUelaxes throughany nev origin, and then making<!==0.Ana mnce the coe&denta of the highest terme are unaltered
by sachtransformation,we must obtain in every casethe samecoeiEc!entsfor a! xy, and and the corves are therefore
similar.If we retain the planea ye and <Kc,and transformthé plane
a~yparatlel to itseM,the section by this plane !s got at once
by writing)?'=cc in the equationof the surface,Binceit Mevident
that it is the same thing whether we write e+ctor e, andthen make ~!=0, or whether we write at once e:=c.
It is easy to prove atgehraicaUy,that the locus of centresof parallel sectionsM the diameter conjugate to their plane,as MgeometncaUyévident.
70. If p', p" be the Mota of the quadratio of Art. 66,their productp' !a==J divided by thé coeindent of p*. Butif wo transformto parallel axes, and consider a radius vector
It is proved(~<<MMMMtB~&o-~<6f<t,p. 112)that if U denotethétenMofhigheatdegreein theequation,andNdenote
(&<t'+(e«-M')y't(at-H')~2(~-<tO)w+2(~-tm)M~a(<&)~,then the equationofthe threeprindpalp!ame<,the centrebeingorigin,is denotedbythedeterminant
0F THE SECONDDMN5E. 47
drawn parallel to the nrst direction,the coenïeientof remauM
unchanged, and the product is proportional to thé new d.
Hence if through two given pointe j&,any parallel chorda
drawn meeting thé snr&tcein pointa J! then the
pmdncts ~JB', jSB.2M"are to each other in a constant
ratio, namely, P': P" where !7', P" are thé resulta of aab-
stituting the co-ordinateeof Aand of B in the givenequation.
71. We shall conclude this chapter by shewing how thetheorems atready deduced from thé discussionof Unes passingthroagh the originmight have been derived by a more generalprocess, such as that employed (CbtttM,Art 150). For aym-metry weuse homogeneousequationswith four variables.
To~nd' <~ pointewhere a given guadric Mmet the line
joining <t00givenJWM!~a:y<'<e',a!"y"<M".Let us take asour unknownquantity the ratio 1 M,in which
the joining line is eut at the point where it meets the quadric,then (Art. 8) the co-ordinatesof that point are proportionalto
and if wesnbstîtatethese values in the equation of the sur&ce,we get for the detenninattomofl m, a $Ma<&'a<M
The coeiBcienisof f and Mt*are easily seen to be thé resoltsof substitatiag in the equation of the surface the co-ordinates
of each of the points,while the coeBSdentof &Kmay be seen
(by Taylor'6 theorem, or otherwise) to be capable of beingwritten in either of the &rm8
Having foundfrom this qnadmtMthe vaines of i' m, sab-
stttatmg each of them in thé values &c.,we find thel+9n
co-ordinates of the pointa where the quadric is met by the
givenline.
MOPERTtEa CMtMOS '[0 ALL SURFACM48
72. If .c'y~'M'be on the aut~ce, then F''=0, and one of
the roots of the last qn&dr&tMis ~=0, wh!oh correspondatothe point a/y'e'M',as evidently ought to be the case. ïn orderthat the second root ahouldako be ~0, we must hâve F='0.
If then the line joining <cy~'«/to a)'y<<a" touch the surfaceat the former point, thé co'ordmatea of the latter must Bat!s~the equation 09.'
and aince <B'y~"o/' may be <M~point on any tangent line
throngh icy~'w' it 6)Uowsthat every such tangent lies in the
plane whoaeequation haa been juat written.
78. If a/y'i!'a/ be not on the surface, and yet the relationF~O be e&tMed, thé quadratic of Art. 71 takea the form
M'P'+ P~7"'=0, which givesvalues of m,equalwith oppositesigna. Hence the line joining thé given pointa is cnt by thesurfaceextemally and !ntem<Jlyin the same ratio that is to
say, is eut harmonically. It &Uowathen that thé locas of
points of harmonie section of radil drawn through aiV~'a)'ia
the polar plane
74. In general if the line joining the two points touch
theaï)'&ce,thequadrat!oofArt.71 muet hâve equal roots,and the co-ordinatesof the two points mnst be connectedbythe relation 4Ï7'P"'='JP*. If the point fcy~' be Ëxed, thiarelation onght to be fulfilledif the other point lie on any of
thé tangent lines which can be drawn throngh it. Hence thecone generated by aU theae tangent linea will hâve for ita
equation 4PT7'=jP*, where
490F THE SECOND DESNSË.
E
75. ?b./M the coo<&'<MMthat <Ae~M tKC+~+'y<!+~ahould touch the auiface given by the general f~MO~Mt.
If a), i:, <a be the co-ordinates of the point of contact,aud X an mdetennmate multiplier, we have (Art. 68)
from which equations, together with ax + + 'yx + 8<o==0, we
have to eliminate a', y, e, <a. But solving for a! y, <, M from
these equations, we have (Higher ~l~m, p. 15)*
It Mthere proved that the coefficientof for example, is the di&-rential of A with regard ton on the supposition that the conatituents of
the determinant A are aU difforent But it is easy to Me that the truedifferentialis double this, aince the detetn~nant bas two symmetticat cm.
etituents each <t.
PBOPERTUN 0F THE SECOND DEGBEË.co
76. The condition that the surface ahould bc touchedby
anyline
is found by eliminating two of the variables between thé equa-tions of the line and of the quadrie, and fbrmmg the conditionthat the resulting quadratic should have equal roots. The
resnit contains the coefficientsof the quadric in thé second
degree, and la aiso a qnadratic function of the determinants
(<~3'), (<r/)) &c. Writing theae (~'), (opy'),&c. the
reault is found to be
If in the condition of the last article ve write a+Xa' for
a, &c., and then form the condition that the equation m X
ahouldhave equal roots, the resolt will be the conditionof this
article muMpUed by the discriminant. For the two planeswHch can be drawn throngh a given line to tonoha quadric,will coincide either if the line touches the quadric or if thé
surface bas a double point.
( &t)>
:q~uun.
E22
CHAPTER V.
CLASSIFICATION0F QUADRIC8.
77. OUBobject in this chapter is thé reduction of the
general equation of the seconddegree to thé mmplest form
of wh!chit is 8Mcept!Me,anAthe classificationof the dMerentsar&eeewMch it ia capable of representing.
Let ua commenceby supposingthe quantitywhich we calledc (Art. 65) nol to be =0. By trMM&nnmgthe equation to
parallel axes through the centre, the coefficientsp, r aremade to vanish, and the equationbecomea
<!a~+~+M'+2~3m.a:+2a.t~+«''==0,
where <? M the result of substituting thé oo-ordinates of the
centre in the equation of the aur&ce. Bemembenng that
where A is the discriminantof the equation.
78. Having by transibnnadon to parallel axes made the
coeSotents of y, <!to vameh,we camnext make the oo-
eSMentsof and xy vanishby ohangmg the directionof the axes, retaining the new origin; and so reduce the
equation to the form
D ia of course <-1 suppose in what Mowe that J) MpoeitiTe.
If it were 0, thé Mr&M would repreaent a cône (A)ft. 63). If it were
negativo, we ehould ontyhave to change all thé ~t~~Min the equation.
Ct.ASMFtCAtrOX OF Qt'ADRICS.52
It is easy to shew from Art. t7 that we hâve constants
enough at our disposai to effect this reduction, but thé method
we shall follow is the same as that adopted, C!M!tcs,p. 141,
namely, to prove that there are certain funetions of the co-
eScients which remain unaltered when we transform from one
rectangular system to another, and by thé help of theso re-
lations to obtain the actual valuea of the new .4, B, C.
Let us suppose that by using thé most general trans<br-
mation which is of the form
which we write for shortness N= & Then if be any constant,
we must have !7+~~=!7+R& And if thé Brst Nde be re-
solvable into factors, M must ako the second. The diserimi-
nants of <7+JÏ~ and of !7+jBjS' must therefore vanish for the
same values of R. But the firat discriminant is
Equating then the coefficients of thé different powem of
to the cofreapondmg coefficients in thé second, we leam that
if the equation be transformed from one set of rectangularaxes to another, we muet have
There Mno dNtcnltyin formingthé eotrespondin~equatioMfor
obliqueeo-ordinstM We thonMthen Mbatitutefor S (<eeArt. t8~<*+y 2yzeo!\ 2« eo~~ !):ycos)',
and proceedingexaettya< m thé text, we ehoutd &nn a euMetnthé t0f&cient<of whiehwould bear ta each other Mtios unalto-edbytmoatbnM&tioï~
CLASStFtCATÏON ÛF QUABRtCA 53
79. The above three equationsat once enableus to trans-form the equation M that the new ?, m, n eMt vanish, sinee
they determine the coei&ctentaof the cuMc equation whm
roots are the new a, b, c. This eubie is then
We give here Caacby'a proof that the roots of this equationare aU real. The proof of a more general theorem, in which
this is mchtded, will be found in ZesMM on jBt~~ Algebra,LeMon XV.
Let the cubic bo written in the form
Let~~Sbethev&htesof~d which make (~-&)(~-e)-f==0,and it is easy to see that the greater of these roots z is greaterthan either b or c, a,!idthat the tess root is lesa than either.tThcn if we mbstttuto in the given cuMc '=' a, it reduces to
and sinee the quantity within the brackets ts a perfect squarein virtue of thé relation (ce –&)(<[–<')=?*, the result of aub-
xtitutton is CMentMy negative. But if we eabstitute
thé result M
which in also a perfect square, and positive. Since them, if
wo substitate ~==00, ~l~a, ~t=~j8, ~=–eo, the résulta are
alternately positive and negative, the equation bas three real
l'oots lying within the Hmitsjust assigned. The three roots are
thé coeStOtents of a~, y*) in thé transformed equation, but
This M the same euMo M that found, Art. 68, M the reader will
easily see ought to be the case.
t We may see tbia cither by aetuaUy aolvingthe equation, or by sub-
stituting successively~soo,~ct,~=c,~=-<o, when we get fesatte
+, +, ehewing that one rcot h jpfatct th<Ht andth<-other leas <han c.
CLASStHCATMS 0F <)UADMCS.M
it is of course afbttriMywMch shall be the coeNcle&tof a!*or of ance we may call whichever axis we please théax!sof.
80. Qnttdrics Me ctasai&edaccording to the signa of theroots of thé precodiag cubic.
I. First, let aU the roota be positive,and thé equation canbe transformedto
The suf&cemakes real intercepta oneach of the three axes,and ifthe interceptsbe <t,},c, it Is easyto see that the equationof the aurfacemay be written in the form
As it H arbitrary whieh axis we take for the axis of a:, we
sappMe the axes so taken that o the intercept on the axis
of x may be the longest, amdo the intercept on the axis of <
may be the shortest.The equation tranafonnedto polar co-ordinatesis
wh!ch (tememhering that cos'a+co~~+coB*'y=l) may be
wrtttenmdtherofthetbrms
<romwhich it is easy to see that a is the maximum and c
the minimumvalue of the radius vector. The sur&M'!s con-
aequentlylunited in every direction,and is calledan ellipaoid.Everyaectionof it is thereforenecessarilyalsoan ellipse. Thoa
the section by any plane <!=.B is-t+~'='l--y)
and we
shall obviouslyceaseto have any real sectionwhen Nis greater
CLASMFtCATÎON0F QUADNC8. 55
than c. The surfacetherefore lies altogetherwithin the planes<='t& SunUartyfor the other axes.
If two of the coefficientsbe equal (forInatamce,a==&),then
aU sectionsby pianes parallel to the plane of a:y are orctes,and the surfaceMone of reoolution,generatedby the revolutionof an ellipseround tte axis major or axis minor,accordingaa
it ts the two greater or the two iem coefScteDtswhieh are
equal. Theee surfaces are aiso eomet!mMcalled thé jpw~teand thé oblatespheroid.
If ail three coeSdenta be equal,the fnu'faceMa aphere.
81. IL Secondiy, let one root of the ouMcbe negative.We may then write the equation in the form
where a ia Bupposedgreater than b, and where the axis of jî
evidently does not meet thé surface in real points. Usingthe polar equation
it is evident that the radius vector meets the sar&ce or not
according as the right-hand mde of the equation is positiveor négative; and that putting it =0, (whieh correspondsto
p =oo) we obtaina systemof radii wMchscparatethe diameterswhichmeet the aurface from those that do not. We obtain
thus the équation of the <Mymp<o<Mcoxe
Sectionsof the snr&oeparallel to the planeof a~ are ellipsesthose paraUelto either of the other two principal p!amesare
hyperbolas. Thé equation of the eUipHcsectionby thé plane0~ M*
z=JBbe!mg-; +~'='t+-t weaeethatweget a real section
whatever be tho value of R, and therotbrothat the surfaceis conttnuoMS.It iacalled the B~X!r&o&)t<~o~(Me~ee~.
If «=&, it ia a surface of revotution.
CLA88ÏFtCATKHt 0F QUADMC8.56
62. III. Thirdly, let two of tho roots be negative, andthe equation may be written
The sections paraUetto two principal planes are hyperbolas,while that parallel to the plane of ta an ellipse
It is evident that this will not be real as long as .B is within
the Umits ±a, but that any plane a:*=jS will meet the surface
in &real section pro~ded that Jï is outalde these limits. No
portion of the ear&ce will then lie between the planes a!=t<t,but the Bm&ee will conaist of two separate portions outaide
these boundary planes. This surface is called thé .B~~Mf
of <«? a~ee< It ia of revolution if &=c.
By considering the surfaces of revolution, the reader can
easily form an idea of the distinction between the two kinds
of hyperboloida. Thus if a common hyperbola revolve round
its transverse axis the surj&tcegenerated will evidently consist of
two separate portions; but if it revolve round the conjugate axis
it will consist but of one portion, and will be a case of thé
hyperboloid of one sheet.
IV. If the three roots of the cubic be negative, the sur&ce
can evidentlybe aatMËedby no realvalues of the co-ordinates.
V. When the &bsolateterm vanishes, we hâve the coneas a Iimitmgcaeeofthe above. FormaI. and IV. thenbecome
wh!ch can be s&tiaRedby no real values of the co-ordinates,while form II. and ÎII. gtvo the equation of the cone in
the&nn
The forma atre&dyenumerated exhaust all the varieties ofcentral sur&eea.
CLA88IFMATMN0F QUADMC8. 67
ByDesCMtes'oraieofsignathisequationhastwopositiveandonenegativeroot,&ndthetefoKMptMentea hypethololdofonetheet.
83. Let us proceednow to the case where we have S'=0.In this case we have seen (Art. 6S) that it !s generally im-
possibleby any change of oripn to make the terms of theSrst degree in the equation to vamah. But it is in generalquite indifferent whether we commence, as in Art. 65, bytransforming to a new origin, and so remove the coeSctentsof a?,y, or whether we fnt, as in thia chapter, transform
to new axes retainmg the sameorigin, and ao reduce the terml
of highest degree to thé form J<B'+J~'+Ce*. When S~O,the nmt transformation being impossiblewe muet commencewith the latter. And since the absolute tenn of the cnbic ofArt. 79Is 8, one of its roots, that ia to aay, one of the three
quantities d, C must in this case =0. The terma of thesecond degree are therefore reducible to the form ~'±J~This M otherwiso evident from thé c<&tsideradonthat o'=00!s thé condttton tbat the tonna of highest dogrcc should bo
~j
CLA.aMFMATMM OP QUAMtCth58
resolvable into two reai or imagmaty retors, in whichcase
they may obviouslybe akc expressedas thé differenceor Mtmof two squarea. In thMway the equation is reducedto thetbna
We can then, by tMasforming to a new origin, make thé co-
eSc!entsof x andy tovanish, but not that cf~, and the equationtakes the form
I. Let /==0. The equation thon doeanot contain and
therefore (Art. 24) représenta a cylinder wh!eh M eU!pticor
hyperbolic,according M and B have the same or dînèrent
signa. Since the terma of the Srst degree are absent fromthe equation the origin ia a centre, but so !s a!ao equallyevery other point on the axis of e, which is called the aieof the cylinder. The posstMIityof the surfacehaving a lino ofcentresMindicatedbyboth numeratorand denominatorvaniahingin the co-ordinatesof the centre,Art. 65 (aeenote p. 42).
If it happenedthat not only but a!so <f=0, thé sur&cewould reduce to two interaecting planes.
IL If f' be not =0, we can by a change of origin makethe abaoluteterm vanish, and reducethe equationto the form
Let aa first supposethe sign of B to be pos!t!ve. In thiscase while the secëons by planes parallel to the planes of aw
or ysare parabolas,thoae parallel to the plane ofa:yare ellipses,and the surfaceis catled the Elliptic jR~fa~o~t~ It evidentlyextends only in one direction, smce the section by any plane<!=<:is ~a?'+~'='-2c< and will not be real oniess the
right-hand side of thé equation is positive. When thereforef' is positive,the surface lies altogether on the negative side
of theplane ofay, and whem is negative,on the positive side.
m. if thé sign of J? be negative, the sections by planesparallel to that of ay are hyperbolas,and the surface is calleda ~p~MtO ~N'aMiM~. This surface extends indefinitelyinboth directions. The section by the plane of a~ is a pair of
right tincs.
CÏ.AB8tFICAT!ON<?-QUADMC8. 59
IV. If .B'=0, that te, if (<Mroota of thé dMcUmmathgcubio
vantsb,the equation takesthe form
but by changmg the axMy and < in their own pkne, and
taking for new co~rdinate planes the plane 9'y+f'<! and a
plane perpendicular to it through thé axis of a:, the equationia brought to the form
wMch(Art. 24) repMsentsa cylinderwhosebaee!s a parabola.fV. If we have a!so ~'=0, /=0, the eqaa~onj~+~~0 0
being resolvable into &ctom would evidently denote a pairof parallel planes.
84. TheaehMdwork~Mdncingtheeqat~onof&p&ra.boloMto the rormA~'+J~'+aJBz=0 M ahortenedby otMervmgthatthe discnnmMtntis an mvMitmt;that t8 to say, a fonction of
the coei&ctentswhich is not altered by transformationof co-ordinates (J~Aef jd~ p. 51). Now the discriminant of
Ac'+I~'+2J&! M mmply~BB', whieh îa therefore equal to
the dtBcnmuMntof the given equation. Amdas and B are
htowD)being the two rootaof the dtscnmmtttmgcâble wMchdo not vanNn, Ma!ao known. The calculationof the d!e<
cnmmant ia &c!Htatodby obsetvmg that !t is in this case a
per&et square (.B<~A<)'~%Mh~,p. 184). Thas let aa take the
exampb
Then thé discriminatingcnMcis X'-6X'ï4X='0 whose rootsare 0, 7, and -2. We have therefore~=7, J?=-8. Thediscriminant in this cMe is (p+2g-3)')*, or putting in the
actualY&tme9~==1, s'=3, )'='3 !a16. Hencewe have ltB*t=16,
Jî= and the reducedequationM~=-,7.~
If we had not availed ourselvesof the diMrim!nant, weshouldhaveproceededM inArt. 68 to find the principalplanesansweringto the roota 0, 7) –3 2ofthe discriminatingcubic, andshouldhavefound
60 CLASMFtCATMM!0F QUADNC8.
Sincethe now co-ordinatesM~ethe perpendicttittrson thèsep!anM,wea]'etotake
from which we can express x, in terms of thé new co-
ordinates,and thé transformed equation becomes
whichfinallytransformedto pandiot axes throngh a new origingivesthe samoredncedequation as before.
If m the preceding example the coefficientsjp, r had been
sotaken as to fulfilthe relatton p + 2~ 8r =0, the discriminantwouldthen vanish, but the reduction could be effected witheven greater facility as the term in a', <!could then bo ex-
pressedin the form
( M)
CHAPTER VI.
PROPERTÏESOf QUADNC8DEDUCEDFROMSPECtAÏ.FOBMSOPTHEIREQUATIONS.
CEt!TBAI, SDMACE8.
85. WE proceed now to give aoaie propertisa of centralw* x
quadUcsderived from the équation + j- + T= 1. This will
includeproperti~ of the hyperMoub Mweti as of the eMpaotdsif we aapposethe signs ofy and of<~tobp mdetennm&te.
The equation of the polar planeof the point a/y'e*(or of the
tangent plane, if that point be onthe air&ce) is (Art. 59)
The perpendicular from the origin on the tangent plane tatherefore(Art. 82) given by the equation
And the angles a, ~8,<ywhichthe perpendicularmakes with théaxesare given by thé equations
as is evident by multiplying thé equation of the tangent plane
byp, ana comparing it with thé form
a; cosa+y cos~S+e<MS<y=p.From the preceding equationswo can abo immediatelyget
an expression for thé petpendicalM*in terras of thé angles it
makes with the axes, "nz.
86. Tofind the <!o?M'<<tOHthat theplane «.B+~y+~-t-S~O 0
shouldtouch<~ NM~~MC.
62 CEKTRA:, SURFACES
Squanngandaddingwenndthat the length of the normaltt
between iC'yV,and any point on it aiys is But If a~ be
taken as the point where the normal meets thé plm of a:y,we
have 0=0, and the last of the three precedingequationsgtveajB='A Hence the length of the mteKept on the normal be-
tween the pointof contactand thé plane of a~ M
88. The anm of thé squaresof the reoprocata of any thiee
rectangular diametera N constant. This follows immediately&omaddmgthe eqoatmna
03COMPGATE DfAMETERS.
89. In like manner the somof the squaresof three perpen-diculan on tangent planes,mntuaHyat right angles, is constant,aeappearafrom adding tbe equations
Hence the locua of thé intersection of three tangent pkneswhich eut at right angles m a sphère aince the aqa&re of ite
distance from thé centre of the surface M equal to the oum
of the squares of the three perpendieulara and therefore to
<+&'+<
CONJUGATE DIAMETEM.
90. The equation of thé diametral plane conjugate to the
diameter drawn to the point icy~' on the surface is
It M therefore parallel to the tangent plane at that point.Sinceany diameter in thé diametral plane is eonjugate to thatdrawnto the point a:y. it Mmanifeetthat when two diametersare conjugate to eaohother, their du'ectMn-coemesare connected
bythé relation
Smce the equation of condition hère given is not altered !f
we write JE~, ~c* for <t', c*, it is evident that two tines
a!* «*wh!ch arc conjugate diameters for any surface + + ==1
are abo conjugate diameters for any aun!!<a'surface
64 CONJC&ATKMAMETEHS.
And by making~= 0 we see inparticular that any surfaceandits asymptotiocônehave commonsystèmeof conjugatedtametera.
FoUowmgthé analogy of methods emptoyed !n the case of
comca we may denote the co-ordinates of any point on thé
ellipsoid, by a cosX, b coa~, o coa~ where /t) y are the
direction-anglesof some line; that Is to say, are anch that
cos*X.+cos*~+coa'f != 1. In this method the two lines answer-
ing to two conjugate diameterB are at right angles to each
other; for writing coset~a coBX,oo8a'==<tcos\ &c., the re-lation last written becomes
cosX eos\' + coa~ cosjtt' + cos~ coaf' = 0.
91. The MtMof the squares o~*? <y~<Mo/' <&feeetmjugate<eM~MMM<e)*<M onstant.
For the squareof thé length of any Bemi-diametera!+y'*+e"is, whenexpMSsedmterma ofX, y,
ia equal to <~+&'+< smce ~t, f, &c. are the direction-
angles of three Unesmutually at right angles.
92. The~xM'a?&!op~eJM&<Meedgesare threeconjugate<e<K~-
<&ame<eMA<M? <WM<aK<volume.For if a~'< a!"y"< &c. be thé extremities of the diameters
the volumeia (Art. 85)
but the value of the last determinant is unity (seenote, p. 20)hence the volumeofthe parallelopipedis abc.
CONJUQAM MAMBTERS. 66
F
If the axes of any central plane sectionbe a' b', and p the
perpendicolar on the parallel tangent plane, then «'atc.For if c' be the semi-diameterto the point of contact,and 6 the
angle it makes with thé volume of the parallelopipedunder
the conjugate ftiametemN',& d M<t'c' cos~, but o' coa~=~.
98. The theoremajust given may abo w!th eaaebe deducedfrom the correepomdmgthéorèmefor conies.
For coMtderany three conjugatediameters a', & c', and letthe plane of &'&'meet the plane of ay in a diameter and let<7be the diameter conjugate to A in the section o' then
wehave ~*+C'-<t"+& there&H~<t''+&+c''=~*+C"~c'
Agam, eince A is in the plane ay, then tf B is the diameter
conjugate to jl in thé section by that plane, the plane con-
jugate to A will be the plane containing B and containingtheaxis c, and <~e*are therefore conjugate diameteraof the samesectionas B, e. Hence we have~*+C*+ o'*=~t J?*+ < and
emce,finally, +J3*=' <~+b', the theorem1sproved. Precîselysinular reasoningproves the theoremabout thé para!Ielop!peds.
We anght further prove theae theoremsby obtaining, as inthe note, p. 52, the relations wh!ch ex!st when the quantity
â + ~+~in obliqueco-ordinatesis transformed to
+ .f.'~+~+.~im ohUqaeco-ordinateaîs irans&nned ',s + %)+*3t
in rectangolar co--ordinates. Thèse relationsare foundto be
o'&V ~<&c'*(l-coa'cos'cos'<'+2 cosXe&s/tcoaf).
The Smt amd tast equationa give thé properties ahettdy ob-tained. The second expressesthat the snmof the sqnaKtSofthe parallelograms formedby three conjugate diameters, takentwo by two, is constant.
~L~~eM~M~M~ej~~M~M~~Me~~J~MM~WM~MeMCM~M~
Let the line makeangles a, ~'ywtth the axes, then the
projection on it of the senu-dMmeterterminating in the pointa:y<t'is a/ ooset+y' coa~+e' 009~,or, hy Art. 90, M
a coexcose + oos~tcos~+ o cosfcos'y.
COtWUQATENAMETEM.66
96. 2~ <MNtof the ~Mfea of F~< three
j~M~wMM~~&~&M~If <~d", d" be the three dtameters, 6" the angleamade
by them with the perpendicularon the plane, the snm of the
squaresof the threa projectionsis <yBin'C+ J" mn'6'+ <F"Mn'O",wMchMconstant, sinced' c<MJ'~+~"co8'+<f"* c<M* Mcon-stantby the last article and <P+ d" + <?"*by Art. 91.
96. ~c~M<eMec<MMO)r~Me<a~Mai <~ e!<MM<MMof threeC<M/t<ediameters.
The equationsof the threetangent planes are
97. 2b.&<M~<<eo<Mtt)M<M5s~<tayj)&MMpttMM~through thecentre.
We can readily form the qttadra~o, whose mots are the
teetpmcab of the squares of the axes, emce we are given thesam anAthe product of these quantities. Let a, /3, 'y be the
angles whicha perpendicnlarto the given planemakes withthe
axes, the intercept by the mr&ceon this perpen~iealar; then
vehave(Arb88)
a?CMMMATE DIAMBTBBS.
F2
This equation may be otherwise obtained from the principles
expMned in thé next article.
98. ï%~o«~ a ~toettradius OJSof a eeMtfa~~!M<&tce cat
«t ~<tMM?draw oneeM<«Mto/'M~tcAORshallbean <KEM.Describe a sphère with 0~ as radms, and let a cone be
drawn having the centre M vertex tmd paernogthrough the
inteneottoa of the surface and the sphère, and let a tangentplane to thé cone be drawn thmugh the radins 0~, then ORwill be an axis of the section by that plane. For in it OB is
equal to the next consécutiveradins (both hemg radii of thé
eameaphere) and is thereforea maximumor minimum; while
the tangent Une at R to the section is perpendicnlarto ON,since it is also in the tangent plane to the sphère. OB îsthereforean axis of the section.
The equation of the cone can at oncebe formed by soh-
tracting one from the other, thé equations
If then any plane iccoM+y cos~S+zcosyhave Mt<udsm
ieng-th =~ ;t must touch thia cône, and the conditionthat itahoaldtonch tt, ia (Art. 86)
CIBCCÏ~B SECTMHfS.68
In like manner we can fmd the axes af any eecttonof &
qMJKCgiven by M equadon of the form
<M~+~*+c~+2~+2aMa!+2<!a~a.l.
The coneof mtemecdoaof this qnadnc withany Bphete
and we see as before, that if be the reciprocalof the squareofan MtiBof the section by the plane a? cosa+ycos~+<!coa'y,thia plane muet touch the cone whMe equationhaa just been
given. The condition that the plane ehonldtouch tMa cone
(Art. 7C)may be written
CIRCULAB 8ECTMNS.
99. We proceed to invesëgate whether it !a possible to
drawa plane which etutHcnt a given e!pso!d in a circle. As
It bas been a!M&dyproved (Art. 69) that all paraM sections
afe stmihr <!nrves,it MBdBc!entto considersectionsmade by
planes through the centre. Imagine that any central section
M a cMe with radins r, and conçoivea concentrioapheredoscnbed with the same radius. Then we have just seen
that
CtBCOtAB fNCFMNS. 69
KpMaentBa cone having the centre for its vertex and pMmmgthrough thé intersection of thé quadric and the spheM. Butif thé smËMesbave a plane sectioncommon,thiaequation must
neceaaanlyKpMSNtttwo planes,wMchcamnottake place anlemthé coefficientof either a! y', or e*vanish. The plane sectionmuet therefore pasa throngh one or other of the three axée.
Suppoaefor examplewe take f c &,the coeCc!entof y vanlahes,and there remaîna
whichreprésentatwo planes of c!reularsection passingthroughtheamsofy.
The two planes are easily constracted by drawing in the
plane of <cea Bemi-dmmeterequat to b. Then the plane con-
taining the axis of y, and either of the seau-diametemwhichcan be M drawn, is a plane of <arcn!arsection.
In like maaner two planes can be drawn throagh eaoh ofthe other axes, but in the case of the eUipaoîdthese planes willbe onagmMy; since we evidentlycannot draw m the planeof
!eya Mml-diametert=c, the teaat send-dt&mete)'in that section
being b; nor, again, in the plane of ~<!a seml-diameter~a,thé greatest in that sectionbeing =&.
In the case of thé hyperboloidof one aheet(~ ie negative,and thé sectionsthrough a are those wMch are real In the
hyperboloidof two sheets where both &' and o"are negative,if we take y'~ -c* (blbeing less than ~), we get the two real'
sections,
These two real planes through the centre do not meet the
surface,but parallel planea do meetit m cirdet. InaRoMes
it will bo observedthat wehaveonly two real central planesof (arcalarsection,the seriesof ptanespMtJlel toeach of whichafbrd two dM9MntSystemsof circularsections.
100. Any two snt&ces whosecoeSctents of y*,e*,dK~r
ontyby a constant, have the sametaïcalar sections. ThtM
~+~*+<&'=!, and(~+F)~+(J?+~)y*+(C+.B)e'~l
70 CÏRCULARSECTMN8.
have the same cirettlar Motions as easily appeam from the
fMmnhtm thé last article.The Mmething appears by throwing the two equationsinto
the form
fromwMohit appears that the dMerenceef the squaresof the
reciprooalsMconstant of the correspondingrad!ivectoresof thetwotnu'&ees. If then in any secëon the radius vectorbe con-
stant, so must aiso the radius vector of the other. The Mmeconmderatîonshews that any plane eute both in sectionshavingthe same axes, since the maximum or minimum valneof theradins vector will in eaoh correspond to the same values of
a, 'y.Circnlarsectionsof a coneare the same as those of a hyper-
boloidto whichit inasymptotic.
101. ~M~<!COcircular <eC<M!M0/' <~p<Mt~<'y<<6)tMlie M <Xeaameaphere.
The equationsof the two planes of section are paralleteachto one of the planes representedby
Now since the equation of two planes agrees with the
equation of two parallel planes aa far as temN of the aecond
degree are concerned, the equation of the two planes muetbe of the form
whereM,repreMntsaomeplane. If themwe snbtract this fromthe equationof the sar&oe, wMch evety point on the aectton
mMta!so)!)ttis(y,~get
wMchrepresentaa sphère.
CïttCULAKSECTIONS. 71
102.ARpMaMeedMmaMe<Mwehave6eeaH!ni!ar. Ifnowwe draw a series of planes pM~Cdto drculNr sectionsthe
<~KmaMM~~lha~pm~dtM~mt~MM~m~meettheaur&ce in amimjBnttelysmatleu'cte. Its point ofMat<M!tMca!Iedam<()M6~M.SomepMpertteacf these pointawill be mentioned afterwards. The co-ordinates of the realnmbilicaare eanty round. We are to draw in thé section,whoM~es~oandc,a8emi-<tiamtter=~~to&tdthe<M~rd!nateaoftheextfemltyofita<~B)ag&te. Nowthe&)]'-mala for comca =<~ -<~ appliedto thiacaMgivesas
There are accordmgly in the case of the eHIpsoidfour Mal
amMIicain the plane of ife, and four ima~naty in each of theotherprincipal planes.
103. It is convenientto add in this placehowin likemannerwe are able to determinethe circularsectionaof the parabobîd
givenby the equation
Considenng a c!renlar section through thé origin, whoseradius ia f, we can see, as in Art. 9$, that tt muet lie in the
BpMrc a
And the coneof intersectionof thia sphèrewith the paraboloidis
Th!a will represent two planes if one of the tenns vanI~M.It will represent two real ptanea, in the case of the eUtptio
pMaMoMt,if we take -=lt for the equation then becomes~i
W= (<~ &*)y*. But in the case of the hyperboHeparaboloidthere is no real ctreular section, aince the same substitution
BBCMUNBAB SENB~tOBa.72
wouldmake the equation of thé two planes t(te the !mag!nMy~nn~+(<t'+t')y'==0.
ïndeed, it can be proved m générât that no aection of-the
hyperboïicparabolold can be a dosed cnrve, for if we take itsmtersectîon with any plane e=&z+)My+M, the pKgeotîonon
thé plme of !s (~+M~+<.) ~chianeeesBM!!y
ahypetbola.
BECTtMNEAROENBt~TORN.
104. We have seen that when the centtal aectton M au
o!I!pseall para!M aectioneare mautar ellipses, and the sec~on
by a tangent plane N an infinitely amaU ainuhr eUipse. In
like maaner whenthe central aaction Ma hyperbola, the MCtton
by any parallel plane ia a mmilarhyperhola, and that by thé
tangent plane rednces Itsetf to a pair of right lines parallel to
the asymptotes «f the central hyperbola. Thns !f thé equationre&rred to any con}agatediametersbe
and. weconsider the section madeby an,. plane pa.ra11el to theplaneof a:e(~=*<3))ïts equationis
And it is évident that thé value ~9==t' redncea the section toa pair of right lines. Such right lines can only exist on the
hyperboloidof oneBheet,Nmoeif we had the equation
the right-hand aide of the equation could not vatush for anyvalue of <. It ia àlao geometncaMyevident that a right !me
c<mnotexist either on an ellipsotd, whieh is a closed snr&oe;nor on a hyperboloid of two sheets, no part ef whieh, as we
)M~BMm~Mq~MïMh~At~wMnMwmJq~~m!~h~pM~dp~M~~M~<myn~~ËM~MNofMmMmgŒm~InteMectthem ail.
BEC'nHNBAB aBNERATOM. 78
10S. Throwing the equationof thé hyperboloidef me oheet
into the form -,A -à-A ~t
!!eaon the surface, and by gMag (tMSmentvalues to wegeta systemof right lines lying in the surface; while, agam, we
get anothersystem by conaHenngthe mtentûctionof the plues
Whttt has been just aa!dmay be st~tedmore generally as
&B<Mn):If ai 'y, S represent four planes, then the equation
tty'~S tepresenta a hyperboloidof one sheet,which may ba
generatedas the locua of the eystemof right lines
Then it in evident that the plane a-~+~y-X'S contaîm
todt, ance ît can be written in either ofthe &nM
74 BECNMNBAB ÛBNERATMa.
the form (a-~8)+JS'(~y-S) can ever be Hent!cat with
(a-~+~y-S) tf XMtdX' are <IMiBMmt.ïn the same
waywesee that both tbe lines
which dffers in thé absolnte term from the equation of the
plane through the nrst Une.
107. We hâve seen that any tangent plane to the hyper-boloid meets the snr&ce in two right Imes intermetingin the
point of contact,and of course touchesthe surface in no other
point. If throngh one of these right lines we draw any o<4ey
plane,wehâve just seen that it will meet the surface in a new
right line, and this new plane will touch the sar&ce in the
pointwherethese two lines intetsect. Convemely,the tangentplane to the mn'&ceat any point on a given right Une in the
surfacewill contain the right line, hnt the tangent plane willin generalbe dînèrent for every point of thé right line. ThcS)take the sar&ce a~'=y~ where the line xy lies on the surface,and and represent planes (though the démonstrationwould
equally hold if they were &nct!ons of any higher degree).Then using the equation of thé tangent plane
MECTïUNEABGENERATOM. 75
and aeekhg thé tangent at the point<o-0,y=0, o=e', we find
.c~'+y~=0, where and are what and beoomeon
eubBtitat!ngth~<a-otd!nate9. Andthispknewm~fu'yM<' vames.
ABtMaM<H<]5Mmtinthee<t9eefthec<)ne. Here everytangent plane meett the mf&ce in two comddentright lines.The tangent plane then at every point of this right HmeMthe
same,and the planetouchéethe sm'&oealong the whole lengthof the line.
And generally, if the equationof a surfacebe of the form
it !s seen precisety, M above, that the tangent plane at everypoint of the line ay is a!=:0.
108. It vas proved (Art. 104) that the two Unesin wMchthe tangent plane cuta a hyperboloidare parallel to thé asymp-totes of the parallel central section; but thèse asymptotesare
evidendy edges of the asymptotiocone to the surface. Hence
every right line which can lie on a hyperboloidis paraUel tosomeone of the edges of thé asymptoticcone. It followsa!sothat no three of them can be parallelto the sameplane, a!nce,if they were, a parallel plane would eut the asymptotioconein three edges, which is impossible, the cone being only ofthe second degree.
109. We hâve soen that any tine of the &-stsystemmeetsall the linea of the second system. Convemely,the surface
may be oonceivedas generated by thé motionof a right linewhich tdw&ysmeets a certain number of Sxed right lines.*
Let us remark in the nrst place, that whenwe are seekingthé surface generated by the motion of a right tine, it iB
necessarythat the motionof the right line shouldbe regolatedby three conditions. In &ct, aince the equationsof a right
Amt)heegenmtedby the motionofa rightHcefil<xt!Ma <t<Mtw&ee. IfMetyt~teMtingUaeiaiNteneetedbyihettexteomecattTeoae,the<ut&MheaNedfK<tM<<!p<tM~.Mnot,!tmM)Ueda<i~wMr)aee.Thehyperboloidofonesheetbelonpto thélatterc!MS)théconetothetanne!
NKTÏMNBAB OBNNtATOBB.7C
!iM mdude four constants, four conditionswotdd abeolatetydeterminethe position of a right line. When we are givenone conditionleM, the position of the line is not detarmiaed,but it Meofor !im!tedthat the line will always lie on a certain
NO'&oe-IoetM,whoMequation can be found aa &Bowa:Writedownthe generaI equationsof a right Une<B Me-t~,y~<M+~-tthea the condtûom of the problem establ!ah three relations
between the constantsM,M,jp, And comMningthese threerelations with the two equations of the right Une, we havenve equations&omwhich we can eliminate the fourquantities
M, y, and the resulting equation in a;, will be thé
equation of the locusreqaifed. Or, again, we may wnte the
equation of the line in the form
then the three oonditionsgive three relationsbetweenthe con-stants a~,y', e') et, -y, and if between thèse we eliminate
a, ~9,y, the reenlting equation in a/, y, <5' ie the equa-tion of the required locus, since a!'y' may be any point on
the line.We Me then that it ia a determinate problem to find the
surface generated by a right line whieh moves oc ae alwaysto meet tkree mxeAright HDes.* For expresang, by Art. 40,the condition that the moveable right line t~aHmeet eachof theSxedlines, weobtainthe three necessaryrelationsbetween
m, q. CteometnctJtyaho we can aee that the motion ofthe line is completdy regnhted by the given conditions. Fora line wouldbe completelydetermined if it were conattamed
to pM8 through a given point and to meet two Sxed lines,aimcewe need only draw phnes throagh the given point andeach of the fixed lines, when thé intersection of these planeswould determinethe line required. If then the point thron~hwhichthe line is to pass, îtsett moves a!onga thM 8xed line,we hâve a determinate aenea of right tine~ the MaemMageofwhich form a am&ce-lccaa.
Or threeBxedcnrveaof anykind.
BECTtUNEABOEMtBATOM. 77
110. Let M then sotve the problem tfNggMtedby the last
article, vis. to find the sanace generatedby a right line whieh
always meets three fixed right Haee. Im order that the work
may be ehortened as moch M possible, let us 6mt examinewhat choiceof axes we must make in order to give the equa-tions of the 6xed right linesthe mmpteatform.
And it occars at once that we onght to take the axes,one
parallel to each of the tbree given right l!net.~ The onlyquestion then !s where the origin eu most symmetncaïtybe
placed. Suppose now that throngh each of the three rightlines we draw planes parallel to the other two, we get thusthtee paireof pandiet planeaforminga paraUetoptped,of whichthé given !mes will be edges. And if through the centre ofthis pattJlebptped we draw lines parallel to thèse edges, weahaUhave the most symmetricalaxes. Let then the equationsof thé threepaire of planesbe
Wee<mMnot do this indeedif the threegivenrightUcMhappenedto be aUpMàUelto thé ttMaeplane. TMoeMewK be een~dered:n thé
BMtArUete. It wilt not oecat whenthe locaah a hyperMdd of one
theet, MeA)t.î08.
mtcnu!n6AaasNBRATOM.78
it Mptesente a central quadrio and ie known ta be a rnledsnr&ce. Tho problem might otherwisebe Mhed thm:
Assumeforthe equationsof thé moveableline
the three conditionsûbtMnedby expMMmgthat thm mtemects
eaohctfthe&mdtmMM'e
thé same equationas befbre.The followingin another general solutionof the same pro-
blem Let the &'st two tmes be the mtefMcttoaof the planesa, ~9;y, S; thenthe equationsof the third canbe expremedin
the form ft'='y~J%, ~'=0y+~. The moveableline, ~aceit meets the mat two lines, can be expressedby two equations«f the form <t=~9, <y==~. Sabetitu~ng ~ese values in the
equations of the tHrd line we find the conditionthat it andthe moveableHmeshould interseot,vîz.
And etonmttm~ Xand between th!a <mdthe eqtmtioDSof the
moveable!me,weget for the equationof thelocus,
111. From the general theory exp!ame(tin Art 105, it is
pMn that the hyperbolioparaboicMmay abo have right !mea
KECnUNZANMNNtATOM. 79
lying ahogethef in the aar&ce. For the equation *=
(Art. 88) M mctaded in the general form ecy'sjSS; and the
snr&ce contaîna the two eyBtemsof right lines
The amt equationmews that every line on the surfacemuet
be para!M to one or other of the two jSxedplanes t o=*0;
and in this respect la the fondamentaldM~Brencebetween rightlineson thé paraboloidand on thé hyperboloid (oeeArt 108).
It !s proved, as in Art. 106, that any line of one tystemmeete every Une of thé other system, while no two Mnesofthe Mme system cammtereect.
We give now thé mvesttgationof the converse problem,vis.
to&td thé aar&cegenerated by a right Ime wh!chalwaysmeet8three SxedlineswMchare all parallel to the aameplane. Letthe plane to whichaUare parallel be taken for the plane of ay,any limewinch meete a three for the axis of e, and let the
axes of x and y be taken parallel to two of the nxed lines.Theo their equations are
wh!ch représenta a hyperbolioparaboloid since the terms of
highest degreebreak: up into tworeal factors.In t!kc manner we might inveet!gatethe surfacegenerated
by a right Unewhieh meets <tpo6xe<ilinesand ie alwaysparallelto a fixedplane. Let it meet the linea
BECnUNBAB OENBBATOB6.~0
whieh taa. hyperboHcparaboloMtancethe tennsof the second
degreebreak cp at tworeal &ctors.
A hyperbolieparaboloid ia the limit of the hyperboloidofone sheet, when the generator in one of its pOMtionamay lie
altogethsr at infinity.We have Mon (Art. 104) that a plane Ma tangent to a
enr&ceof the seconddegree when it meeta it in two Mal or
mu~tMry lines; and (Art. 83) that a paraboloidM met bythe plane at in&Mtyin two real or irmaginarylines. Hencea paraboloid!aalwaystouchedby the planeat m&u<y.
112. JRw ~< ~MMNMM~M~ <0Me ~0!t <<<NK~Mt
te&M~M!~to <~eother ~<<eMin a CMM&M<<M~<MWM)M<!Mt<M.For throngh thé four lines amdthrough any Une whieh
meets them aH we can draw four pJaaes; and thereforeanyother Mne which meete the four linea will be divided in a
constant anhannonioratio (Art. 88).
Conversely,if two mon-mt'BmectmgImteeare divided&Mnc-
~~ttc~~maaonesofpomt~thatmtoaay, so that the
amhaaaonio ratio of any four points on one line Mequal tothat of the cortespomdmgpoints on the other; then the lines
joining cometp<mdin~pomtB will be gemeratomof a hyper-boloid of one aheet.
Let the two given lines be <t,~8;y, & Let any nxed linewhieh meets them both bo e<='X'/9, 'y=~; t~en<'D orderthat any other line ft~A~, v='~ shoulddividethem homo-
graphtcaUy,we musthave (Ci~MM,Art. M) '= aad if we
aUB)?AOZ80F BEVOMJTMN. 81
Q
eliminateX betweenthe equationea <= \'<y =~o, the MMtttia \y=~'a&
118. In the case of the hyperbolicparaboloid any three
right lines of one system eut all the right lines of the other
in a constant ratio. For mncethe generators are all parallelto the same plane, we c<mdraw through any three generators
parallels to that plane, and all right lines which meet three
paraHet planes are cat by them in a constant ratio.
ConvorMiy,if two finitenon-intersecting lines be &v!ded,each into thé same nnmber of equal parts, the lines joiningcorreapondtngpoints will be generators of a hyperbolicpara-boloid. By doing this with threads, the form of this surfaoecan be readity exMMtedto the eye.
To prove this du'ectty, let the line wHch joins two con'e-
sponding exttemities of the given lines be the axis of <; letthe axes of a; and y be taken parallel to the given lines, and
let the planeof xy be half-waybetweenthem. Let the lengthsof the given lines be <tand o, then the co-ordinates of two
correspondingpoints are
BUttB'AOESOf MVOLCTNN.
114. Let it be required to Snd the condMona that the
gMeï<q~ttcmBho~dïepresont&aurfa~of révolution. Inthiacasethe eqa~om can be redoced(eeep. M), if thé eai&ce
be central, to the&rm ~)+~ti'~='iÏ<
and if thé surface
SUBFACESOF MVOMJTÏON.M82
be non-centralto the &nn + e" ln oither CMCthenor <r e
when the Mghest tenna are trMM~btmedso as to becomethesumof aq~Meaof three rectaagniM'co-ordiatte~ the coefMentsof two of thosoaqoMesare eqnal. It wouldappeaf then thatthé required eondition would be at once obt<unedby formingthe conditionthat the diseriminatingcnMcehonMhave eqnatfootB. Sincehowever the rootsof the diecaiminatingcâble are
alwayspoNtive, its discriminantcan be expressedM the sumof squares(aeeJB~~ ~e&M, p. 184),and will notvmieh (theooeBSclentaof the given equation being snpposeAto be real)nn!eN<MMconditionsare AtMHedwh!chcan be obtainedmore
easHyby the followingproceas. We want to <!ndwhethertt Mposaibleso to transformthe equationas to have
the threeformer of whîch are mchtded in thé three latter. In
&epMeeBtcaM<htmtheaeIatterth]'eeeqnatt<HMar6
SoMng &r from each of theae equattons we see that thereductionu impossibleantessthe coe&c!entsof the given eqn~tion be eonnectedby the two retati&as
ThM ia to my, the reciprocal equation MBMhMident!e<t!ty.
SURFACES0F BEVOMTTMN. 88
ptUHMQ22
If these relatîoMbe MNted and if we subatitute any of thèsecommon~tdnesfor in the fuBettom
and âmmeethé plane<?'='0 représentaa planeperpendicub)*to thé
axis of revolutionof the surface,it followsthat + + =0t <M n
representsa planeperpendicularto that axis.IntheapecMt~se~epethecommonTalueevamBhwMch
have been just found for X, the highest terma in the given
equation fbnn a pe!'6Mtsquare, and the equation representaeither a paraboKocylinder or two parallel ptamee(aee IV.and V., p. 69). Thèse are limiting cases of sar&ces of re*
volution, the axis of revolution in the latter case being anyline perpendicularto both planes. The paraboHecylinder Mthe limit of thesnr&oegeneratedby the revolution of an eMtpaeround its trMmvemeaxis, when that axis passesto infmity.
116. If one of the quantities ?, m, ? vacish, the surface
cannot be of revolutiontmleasa secondatso vMuah. Supposethat wehave i <mdmbothee0, the preceding conditionsbecome
116. Thé precedingtheory might a!sobe obtainedfrom theconsiderationthat in a enr&ce of revolution the proMem of
84 MCt.
findingthe pruMipaÏplues becomesinde&ute. For tanceeverysection perpendicalarto the axia of revolutionMa M!'e!e,anyeystemof pMMIetchordeof one of theae cirotesia biMctedbythe plane p<Mts!ngthrough the axis of revointion,and throughthé diameter of the circle perpendicularto the chants,a planewhioh ia perpendtoobr to the chotth. It Mewo that «M~yplane through theaxis of revolutionMa principalplane. Nowthe chordswhichare perpendicularto these dtametndplanesare
given (tteep. 45)by the equations
(<JB)<c+My+<)M!!=0,<M+(&)y+&!==0, Ma!+~+(c-jB)a'-0,whieh when B is one of the roots of the discximinatingcubic
represent three planesmeeting in oneof the right Imeerequired.Thé problem then wiû not becomeindetennm&tetmlessthèse
equationaaU represent thé same plane, forwhich we have theco!'d!t!on9
which expanded are the aame as the condMone &und ah'eady.
MCI.
117. We ahall conctude this ohapter by a few exmmples of
the application of AJgebraIc G~ometty to thé M<)M<~a<t!wof
Loci.
]~l.To&ad~h<M~apeintwhoM<hm~<~tM~<~tw(t
given n~n-inteMecting ri{;htUnesare eqneLIf the equations «f the NM<are written in their general fom the Mttt-
tion of this la obtalned inmMdiatety by the formula of Art. 14. We may
get the tesatt in simple fMm by taking for the Mti<of «te ehottmt
dhtMMiebetween the two Unes, and ehooting for the other Met the Unes
MMedag the angle between the projections en their plane of the given
Unes, then their eq~ttene me of the &tm
MM. 86
ïf thé ehmtMtdhttMMh<dbeeot<t«tehoato- ta <givenmtto,theheMwouMhtwbeen
wMchMpMMBtoa hyperboloidof oneaheet.
Ex.2. To &td the !oetM«f the iaiddtt points of «tt !iNMpttaNet tea &te<tplaneand tennintted by two aen-inteNeethtgM"<
MM the plane <ec0 pm<d to the &<edplane, and the phM <- 0,M!nthé het examp!~poMRetto the twolinesand eqnidhtmt&mathemJthentheeTMttioM~theHawMe
ThehxMh then etMtenttythe right HnewMehfa the mteKeetionof the
p!mM 1c-_n<t'*0, !ty'*(m+M')<tt(<
Ex. 3. To Snd the Mt&ee of revolutlon generated by < right Une
tmntng r<Mmda &Md Mtit whieh it does not MteKMt.têt the Cxed Une be the aiis of< and let any portion of the other be
<OM+M, yt~o~of. Then Mneeany point of the tMctvm~ line de-M)tbMa cMe in a plane pantM to thtt of ay, it follows that the ~h~
ofjt'~y'MtheMaae~erety point in Mehttplane Mctioa,MnHt!<pMnthat the comtaat w!ae MpreMed in terms of < h (Me+ a)*+ (<?')! <t')*.HeMe the equation of the tpqaired Mï&M is
whiohMpteMntt a hyperboloid of revolutionof one aheet.
Et. 4. Twolines paMiagthteagh thé origin move each in a ihed plane,remaining perpendicular to eaeh other, to Nad thé Mt&Mie(nee<M<~tya
cane) ~enettted by a right line, aho pttMin~throagh the origin perpen-dieattf to thé other two.
Let the Ahection-angtM of the petpend!eah)m to the &ted planes be
«,&,<} «', f, MtAlet those af the variable line be a, 'yt then thedheethm-eoeinetof thé intMMcHomwith the Bxed plane*, of a plane pet-pendietthr to the vatiaMe tiae, will be proporëonal to (Art. M)
eM~eoM-cM~6M&, eM~e<x«-cM«eMe, eoeaeMt-cM~eostt.
eot~cote'-ewyeotf, «M'ycota*-CMecote', eos"cM~-e<M~eoto',and thé condition that the<e thoaM be perpendieuIar to each other la
86 LM.
Ex. a. Two ptaoea matMHy petpendicatM paM Meh through <t&MdMne! to ~nd the «Ut&ee generated by thett !)M of imtetMetifm.
Ttk~t~MMMinE~ï. 'nMnQwMmttoMofthenhaMtM
which MpKMntt a hyperboloid of one eheet.If the tme<intetseet, !a which case e o, the !oew teducee to a eone.
Ex. C, To M the tocm of a point, whence three tangent lines,mutnaNy
at right angtM, aenbe drawn to the quadrio ~:+If the equation weM ttMM&atMdM that theee HnMthoald beemne thé
Mes of e<w)tdin<t<eo,the equation of the tangent eone would take the form
~t~t JRM + Oty 0, ainee these thfM Hne~ <tNedgM of the cone. Batthe antMMËMmedequation of the taB~entcone i~.Me Art. 74,
And we have Men (Art. 78) that if fhia equation be ttOMtM!Md to anyMCttngNht syatem «fMMttheBnmcftheeoeNeienttofz* M)d<~will
be constant. We have only then to MpreM the condition that this aum
thoald ~<nMt,~hea 'm obtain the e<p<ati<tnof ttte required locus, viih
The &n<w!!t);method may be u<ed in general to Bnd the équation ef
thé «me whose ~ertex M ~yz'M', and bau the interMetion of any two
eui&M)eaO, 8m1~tote)ne~eqMt))Mt<Mf<)',w+~!<!M'y,yt\y',&e., and let thé Mtwttt be
MC!. 87
then thé reault of eMmhMt!')g between theae eq<MtttoMwill be the eqatt.tion of the required coM. For the tMie~ ~'Me thé line joiaing o~ to
<y)w meete the Mt&MieF are sot &om the Ûntt of theae two eqwt!o!MtthoM where the Mme Hne meet< the eat&ee are got &om thé Meond:1and when the eliminsnt of the two eqaattom TanMtet they tMtvea oommon
Mot, or the point o~~ Hex on a line pMNBg through o'V)! and meetingthé interaection of the Mt&eee.
Ex. 8. To &ad the equation of the eone whoae vertex is the eemtfe of
an elIipsoidand bMe the Mcdon made by the polar ofany point .<y~.
t* o* j~*Ex. 9. To <tndthe toem of points<mthe quadrio +s y"
the
aomMbat wMehinteMeetthe monnalat the point .t~f.~M. ThepotnteKqaitedM'etheintBteeettonofthemt&M~ththeeone
Bx. 10. Te &td the !oca< of the potes of the tangent ptanea of one
quadrio with respect to another.
We haw only to express the condition that the polar of jt-Vif~, utith
regard to the second quadric, ahould touch the BMt, and have <hete&<K
MtlytomMtato~, ~<br<t,y,<iettMe<mditt<!Ogiifea
Art. M. The !ocMi< thet~bte a quadrio.
Eic. H. To Cad the cone geneKtted by perpendie~me eMcted at thevertex of a given cone to ita sevemi tangent planes.
Let thé cone be JS<*+ My' + jM* 0, and any tangent plane is
J~it!<+J~'y+JM:"0, the perpendieolM to which through the origin
7~7'* ~7°')~' we eaU thé eontmon wtne we hâveiaC a
If then we eaU the commoa valne p, aro have
o)'y' ~Mbstitutmgwhich~d)tMtnJ'tJ<<
diMppeMa, md we have-y
+~+ y"
0. The &<nmof the eqn&Hoa
ahewsthat the relation hetween the conea ta McqMoca!,and that thé edgetof the CMt are petpendieubr to the tw~eat plane to the second. It eau
eaOlybe seon that this is a partioalar case of the blet eMMpb.ïfthe equatlon of the cône be given !n thé form
<M'+~'+<!Z'+&~t+S<M!+~!y'-0,
the equation of the KdpMCtd coae will be the same M that of the re-
ciprocal enrve in plane geometry, ~i)t
88 LOCI.
Ex. M. A line move< about 80 that three nxed pointa on !t move onBzed p!anee te ned the locus of any other point on it.
let the eo~o-dioate*of the IceM point P be a, ~) and let the three&ted planes be tahen for eo-o'dinate planee meeda~ the Une in pointa
B, C. Then it h eaay to see that thé eo-ofdiaatea of are 0,.tB Ac
whoro the ratioe s3B PB, d C sPC aro ânown. Eac-?B~' p7*
~J?: ~C':J"C me hnown. Ex-
pteMin~ then, by Art. M, that thé diatance J*~ !Beom)<Mt,thé locusM at once fo<!ndto be an eUipMH.
Ex. 13. ~t and 0 are two &xed pointa, the latter being on the mt&Mt
of a sphere. Let the line joining any other point D on the sphefe to ~[
meet the aphere agaim ia J')'. Then if on OD a portion OJP be taken
-y.Bndthetocmoff. [Sir W. R. HamUton].We bave ~D'. ~O* t OJ3' S~O.OJ') co~~OD. But ~D ~-iea
inveKety as the radius vector of the locus, and OD il given, by the équa-tion of the aphere, in terma of the angles it makes with Oxeda.M~ Thusthé tceus il eMiIy oeen to be a qu&dtieof which0 i<the centre.
Ex. 14. A plane pMie< throngh a nMd line, and the Unes in whieh
it meeta two Kxed planeo are joined by plana each to a hed point; &ndLthe Mr&ce generated by the tine of inteNection of the latter two planes.
Ex. 16. The four faces of tetrahedron paM each tbrough a &Mdpoint.Find the loca* of the -vertex if the three edges which do not paeathMnghit move each in a &[ed plane.
The locus M in general a surface of thé third degree having the inter-
Beetion of the three ph~nMfor a double point. It redaoes to a cone of
thé second degree when the four nxed points lie in one plane.
Ex. 16. Find the locua of the vertex of a tetrahedron, if the three
edge) whieh paMthMttgh thtt vertex eaohpass through <t&[edpoint, if the
oppoeite face aho pMS thmugh a nxed point and the three other verticea
move in &xedplanes.
Ex. 17. A plane pamet throngh a nxed point, and the pointa where
it meeta three nxed Hneaare joined by ptamet, each to one of thtee other
Bxed !ine< Bnd the Ioe<Mof the inteNeetion of the joining planes.
Ex. 18. The aides of a polygon in space pass thtough nxed pointa, and
aU the vefttceabnt one move in &[ed ptanes! Ond the carve Io<)Mof the
remaining vertex..
E~ 19. AM the aides of a polygon but one paM thtough nited pointathe eïtjenutiM of the &ee aide =ove on mM&lines, and all thé other
wrdeea on &[ed ptanM, nnd the NM&eegenerated by the &eeaide.
( 89 )
CHAPTER VII.
METHOD8OPABMDOEDNOTATION.
118. WE shall in this chapter give an accoant of some ofthose propertiesof quadricswhiehare most simply derivedbyméthodeanalogous to those explained m Chap. XïV.of théj~ee~Meon ûb)!«w. In order to economize space we eha!t
occasMnaJIysnppMMsnch details as we think ought to présentno di~5ca!tyto an intelligent reader. In particular we !eave
it to the reader to showthat the whole theory of BeciprocalPolars)asexplained in Chap.xv. of thé <XMMca,appKesequallyto spaceof threedimensMM,thé polars being taken with respectto any quadric. We ahallthus dispense with thé neceasityof
giving separatepMo&of a theoremand of its reciprocaL Inthe methodof ReciprocatPo!ais it will be observedthat a pointcorrespondsto a plane and vice<< and that to a lino (join-mg two points) correspondsa line (the intersection of two
planes). In order to show what corresponds to a carve in
space we shaU anticipate a little of the theory of ourveeofdouble cnrvatuK to be explainedhereafter.
H9. A curve in space may be considered as a series of
points in space1, S, 8, &c. arranged according to a certainlaw.If each point be joined to its next consecntîve,we shaUhavea series of lines 12, 3~ 84,&c., eadi line being a tangent tothé giveneurve. The assemblageof these lines formea snr&oe)and a <&M&)p<esni&ce (seenote, p. 7&)Nnce any Une18intemectsthe consécutiveline 38. Again, if we considerthe
phnes 128, 284, 84S, &c. containing every three consécutive
points, we sha!l bave a seriesof planes which are called the
OMt«htMtyplam of the given carve, and which are tangentplanes to the developablegenerated by its tangents. Nowwhen we reciprocate, it is ptain that to the series of points,lines, and planes,will corresponda series of planes, lines, and
MtSTBOM 0F ABMMBD NOTATION.90
points,and thus that thé reciprocalof a sériesof pointsforminga curve in space will be a series of planestouchinga develop-able. If the curve in spacelies all in one plane, the recipïocalp!aneswill all passthrough one point, andwill be tangentplanesto ac<MM.
Thus the series of points commonto two surfacesiorms acnrve. Reciprocailythe mies of tangentplanescommonto twosar&eestouchesa developablewhichenvelopesbothsnr&cea.
The degree of any Bur&cebeing measatedby the nnmberof pointem whichan arhitrary tine meeta it, the degree of the
sm'SMerociproealto a given one M the Bameas the numberof tangent pitmeswhich can be drawn to the originalsurface
throngh an M'bitrary right !me. The reciprocalof a quadricia a quadric,aince it may be eeailydeduced,&om Art. 76,thatbnt two tangent planes can be drawn to the quadric tbroughan arbitrary Mne. The sametheorem is provedby forming,as
at p. 87, the actnal equation of the locua of the polar with
respect to the quadrio of the tangent planeato another, which
equationioat onceproved to be of the seconddegree.
120. Let now F and F represent any two quadrics, then
F+XP~ represents a quadric passing throngh eM~y pointcommonto C~and F, and if X be indeterminateit representsa series of quadrics having a commoncurve of intersection.Since nine points determinea quadrio (Art. S4), !7+\F is themost general équation of the qnadrîo passing throngh eightgivenpoints (seeJ3%f P&MMCM<w<,p. 21). For if !7andbe two qnadrics,each passingthrough the eight points, Ï7+XF
representsa quadric abo passing throngh the eight points,andthe constantX can be so determinedthat thé snr&oeahallpaasthrough any ninth point, and can in this way be made to coin-oide with any given quadrio through the eight points. ItibUowsthen that all quadrica which pass through eight pointshave besidesa whole seriesof commonpoints, forminga com-mon curve of intersection and reciprocatly,that ail quadricswhich touch eight given planes havea wholesériesof common
tangent planes determininga 6xed developablewhichenvelopesthe wholeseriesof surfacestouchingthe eight Sxedplanes.
MBTaOM 0F ABNMBD NOTAMON. 91
It M evident a!so that the problemto describea quadrieihrottgh nine points may becomeindeterminate. For if théninth point lie any where on the cmrvewhich,as we bave jnstseen,is determinedby the eighth&tedpoint, then ew~ qaadrio
passingthrough the eight nxed points will pasa throngh the
ninthpoint, and it is necesMttythat weBhoaldbe givena math
point,Kc<on thia curve, in order to ho able to determinethesurface. Thus if U and Ybe two quadricathrongh the eight
pointawedéterminethe snr<aceby mbatitntingtheco-ordinatesof the ninth point in P'+\r'=o; but if these co-ordintteamake !7'=0, F~O, this substitutiondoesnot eMbte us to de-
termine
181. Given seven points [or tangent p!anes] commonto aseries of qnadrics, then an eighth point [or tangent plane]commonto the whole system is determined.
For let F; W be three quadrics,each of which paMes
through thé seven pointa, thon !7+XF+/tW may represent<Btyquadric which pâmesthrough them; for the constantsX,
may be so determined that thé Bnr&oesha!l naM through
any two other pointa, and may in this way be made to coin-
cide with any given quadrio thron~h the seven points. But
C'+~F+~f representsa surfacepassingthrongh all pointacommonto P, f, W, amdsince ~ete intersect in eight points,it followsthat there is a point, in additionto the seven given,wMchis commonto thé wholesystemof sar&ces.
We see thus that though it waa proved in thé laat article
that eight points in ~MeM~determinea earve of doubleenrva~
ture commonto a systemof quadrics,it is ~MMMethat they
may not. For we have jnst seen thtt there is a pM'tieataroaae
in whichto be given eight points ieonly équivalentto being
givenseven. When we say thereforethat a quadric is detei'-
minedby nine points, and that the InteMectionof two quadricais detenninedby eight points, tt is assmnedthat the nine or
eight points are per&ctly anrestncted in position.*
'nieMaderwhobasttmMedJa%'A<fPlan Ct<twe<,Arts.2S-S7,wUthaveM diCcu! in deve!ep!agthec<a'Mqx)adingtheoryforMt&ceeef
any degree.ThMif we<MgivenonetM<thanthe numberofpoints
92 MNTHOMOFABMDOKCNOTAttON.
122. ïfa<y<tem«fqMdr!MheM If a ey~temof qatfhJMbe t«-t(eem)ao!teutveef!nteMection,thatMtibed!o the Mmedevelopeble,fato say,if theyhave~ghtpoint* that fa to My,if theyhavee)f;htia eommen,thépolarplaneof any «Mmnontangentptme~thetecua&M<tpointpM<etthrougha Cxed of thé poleof.a <Mdplaneis a
)f!ght!iM. t~httine.
For if P and Q be the polar planes of a SxeApoint with
regard to Ï7 and Vreapectively,then F+X$ is the polar of
the Mme point with respect to Ï7+XP.In particular, the locus of the. centre: of all quadrics in-
acribed m the aame developable,or tonchmgthe eameeight
planes, is a right Jine.
128. If a systemof quadricspMs tbrough a commoncarveof intersection [or be tnscnbed in a commondevetop~bte],the
potMeof a fixedHuegenerate a hyperboloidof oneaheet.
Let thé polaMoftwo pointa in the line be f+X~, JP'+X6*,then it !a evident that their intersecttonlies on the hyper-boloid ~=F'e.
124. If a systempaM through a commoncurve, the locua
of the pole of a SxeAplane is a curve m space of the third
degree. For eltminating betweenP+ F' + f" +\Q"we get the f~ytttemof determinants
which represents & cnrve of the third degree. For the inter-
aecdoa of the aar&cea represented by ~=.jP' P~"=jP"~N a curve of the fourth degree, but this mdades the righttuM JP~, which is not part of the mtetaecdon of P~"=.P"Ç,
necemary to determine a MNf&eeof thé degree, we are given a <e)~eaof pointa &a'ming a carte through which the Mtt&ee must pm<) and ifwe m ~ttea two leas than the number of points neeeMMyto detenainethe Mt&ee, then iM are given a oertain nombe)' of other pointa [MmetyM many as will make the enth'e number up to <t*jthMtgh whMt thesurface mMt aho pMe.
METHOM or ABBÎMED NOTAMON. ?
jP'=f"Q'. There m therefore only &curve of the third
degree commonto ait three.
BectprooaUy,if a aystembe inscribedin the same develop-able,thé polar ofaBxed point envelopesthé developablewhiehMthe reciprocalof a carve ofthe third degree.
126.Given<wvenpointeia a Givenseventangentplanestoquadric,thepolarplaneof &Md a quadric,thepcteof<&cedplanepointpaMMtbrougha &tedpoint. OtovMina t)tedplane.
For evidently the polar of a Sxed point with regard to
!7+A.r+/<~wiU be of theform P+\~+~ and will there-forepassthrough a nxed po!nt*
126. Since the discriminantcontams the coefficientsin thefourth degree, it followsthat we have a Mquadraticequationto 6o!veto détermine X, in order that C~+~~may representa cone,and thereforethat ~M~ theM!te!'<ec<hMt~*<Mo<~«M~Ma
j~Mf co)M<may &cdMcWM. The veruces of thèse cones aredeterminedby the intemecdonof the four planes,
where !s one of the Mots of the Mquadraticjust K&ned
to; and they are given as the four pointecommonto the
series of detemHmmts,
There are four points whoee polam are the aame with respectto aH qattdnce paasing thtongh a common curve cf !nte)'Be<t!on,
Dr. HeMe hM derived from tMt theorem a eeMttaetion for the
quadrio pM<it)g through nine gi'Ma pointa. <~<Kt,VoL xxiv. p. 3e.
<~MttM<~ <M<tJC~MM~<tt)tM<M<!<J<M<wa<,Vo!. IV. p. 44. Sea <!<?
Mme &ttther <tevelopmenta of the eame problem by Mr. TewMend. ?.,VoL tv. p. Mt.
M MEMMM0F ABMDQBBNOTATION.
nameLy)the vertices of the four ooneejust re&rMd to. Forto expreœ the conditions that
should KpMsent the same plane, we Sad the very Mme setof determinants. In like manner there are four planes whoM
potes are the aame with respect to a set of qcadncs inscribedin the aarnedevelopable.
127. As in the case of C~ce (see Art. 298), if the two
quadrica !7 and r touch each other, the Mquadraticm X willhave equal roots. TMa may he most easilyproved by takingthe origin at the point of contact, and the tamgent plane forthé co-ordinate plane z. Then for both the quadrica we ahallhave <=0,=0, y==0, and subatitating thèse values in thé
dMomunamt,p. 41, the Mqnadratic becomes
which haa two equàl roots. The condition then that two
quadrica ahould touch is obtained by formingthe discriminant
ofthebiqnMb'atMÎn\.Im général, it is evident that the ratios of the ooeBMento
~f~MtM~M~~MmX~nUbeu~MM~swMhK~Md~~Mpair of quadrics.
128. It M to be remarked that when two aarEMestonch,the point of contact ia a double point on their curve of in-tefseciioN.
Ingeno'a~twostu'&cesof the<tt'*<mdK'*deg]'eesr6-
spectivdy interaect in a cnrve of the ata" degree for anyplane mcets the aotB~oesin two carvea which intersect m ~a
points. And at each point of the cnrve of !ntemect!anthereia a singletangent line, namely, the intersectionof the tangentplanes at that point to the two sur&ces. For any plane drawn
through thiaune meets the soj'&cesin two curves whichtoachsnch a plane therefore passea throagh two coincident pointaof the eorve of intersection.
METHOM OF ABBtDGB!) NOTATMN. 86
But if theaurfacestouch,then epoy plane through the pointof contact meetf)them in two curvea whieh touch, and everysaob plane therefore pasaeathrough two com~deat points ofthe curve of intemectioa. The point of contact is therefore
a doublepoint on this oarve.And we can ahow that, as in plane cafVM,there are two
tangents at thé double point. For there are two directionsin the commontangent planeto the surfaces,any planethrougheither of which meetsthe snr&cesin curveahaving three pointsin common.
Tftke the tangent plane for the plane of ay, and let the
equationsof the aur&ceBbe
then any plane y=/M! eute the eortacea in curvea whieh osca-
tate (see C~&a, p. 206), if
The eame may be otherwiseproved thus. It will be provedhereafter precisoly as at J3~ Plane C«n)M,p. 27, that if
the equation of a surface be «,+M,+«,+&c.'=0, thé originwill be on the surface,and «, w!Uinclude all the right Mmeawhichmeet the surface in two consécutivepointaat the <;ïigin,while if M,la identtically0, the mrface haa the origin for adouMe point, and <~inctades all the right lines whieh meetthe surface in three consecativepoints. Now in the case weare considering,hy mbtMcting one equation from the other,we get a surface throagh the carveof intersection,viz.
in which sar&ce the ongin is a double point, and the twolines just written are two tnMttvhich meet the sm&ce inthree consecutivepoints.
129. When theaolinescoincidethere is a coq) or etationarypoint (eee J%f~' F&MteC~t~M,p. 28) on the carve of inter-
9C MSTBODS0F ABNMEDNOTA'nON.
section. We ehall caU thé contact in this case stationtuycontact. The conditionthat this ehouldbe the case, the axes
beiag aMumedas above, is
Now if we compare the biqaadratiofor X, given Art. 127,remembering abo that in the form we are now working with,we have )'=' we shalt see that wKenthis condition is
fulfilled, three roots of the biquadraticbecomeequal to 1.
The conditionsthen for atationarycontactare roundby formingthe condition8that the MqMtdrat!oaho~d have three equalroots, vu! ~a'0, y==0, jS and T being the two mYarMmteof the Mqnadr&dc.
130. Smoe the condition that a quadric abonld touch a
plane ~Art. 7~) involves the coefficientsin the third degree,it follows that of a syatem of quadrics pMMngthrongh acommon curve, three can he drawn to touch a given plane,and reciproca1ly,that of syateminscribedin the same develop-able, three can be described through a given point.
It is obviousthat in the former case one can be descnbed
through a given point, and in the latter, one to touch a given
plane.Im either case, two can be deacribedto toucha given line;
for the condition that a quadric abouldtouch a right Une
(Art. 76) involvesthe coeme!ent8of the quadrioin thé second
degree.
181. It is atso evident geometrically,that only three quad-rios of a aystem having a commoncurve can be drawn to
touch a given plane. For thMplane meetathe commoncurvein four points,through whichthe sectionby that plane of everysurface of the system must paas. Now, s!noea tangent planemeets a quadricm two right lines,real or imaginary,(Art. 104)these right !!neain this case can be only someone of the three
paire ofright lineswhich can be drawnthrongh the four pointa.The points of contact whieh are thé pointawherethe tirnesofeach pair interaect, are (<XMt<M,p. 193)each the pôle of theUne joHungthe other two with regard to any cornopaaaing
METaOMOPABKtMtEDNOTATION. 97
H
through the fourpomts. Henee, (p.M) if the verticeaof one
of thé four conesof the Systembe joined to the three points,the joining lines are conjugatediametera of this cone.
132. A system of sar&ces having the eame centre and
commoncircularsecttonsmay be regarded M a particular case
of a system having a commoncurve; for their equation haa
been proved (Art. 100) to be of the form j8'+\(i);*+y'+~.And ince <+~*+<s' repfeaenta a cône, it appears that the
common centre !s one of the four vertices of cones of the
system. Moreover,any three coqagate dtameters of the ima-
ginary cone a!*+~+<==0 are at right angles to each other,sincethis equationrepresentsan innmtely amall sphere. Hence<A~eCMtceKt~Mand eoKcye~M%t«K~t<MMM <~«'~tM to &MMAa given plane, and the MtM~MNM~<Xethree ~OMt<t<j)f<!ON<ac<<othe <ieH<t~are MM<!«t~at a~XM.
133. If two quadrics touch in two points, their carve of
intersection, wh!ohin the general case is a carve of doublecarvatore of théfourth degree,breaksap into two planeoon!cs.For if we draw any plane throngh thé two points of contactand throngh any point of their mtemection, this plane willmeet the quadricsm sectionshaYmgthree points common,and
havingcommonabo the twotangents at the points of contact;thèse seodonamust therefore De identical. The equations ofthe quadricswillthen be of the fonn j8'=0, <8'Z~f'=0, whereL and representthe planes of sect!on. It !s proved in likemanner that the earfacesare envelopedby two commoncônesof the seconddegree. For take the point where thé mteï~-sectionof the two given commontangent planes !aeut by anyother commontangent plane; then the coneshaving this pointfor vertex, and envelopingeach snr&ce, have common three
tangent planesand two edges,and are there&MIdendca!. The
Kciprocah of a pair of quadrics having double contact will
maeestly he a pair of qnadricshaving double contact, andthe two planes of intersectionof the one pair will correspondto the verticee of commontangent cones to the other pair.Any point on the line Z~tf will have the same polar with
98 MBTHOM 0F ABMMN) NOTATION.
regard to aU sur&cesof the System~+~jM. For if P he
thé polar ofjS,&e polar of~\MfwHl in general he
P+~(L'Jf+J'<Af), whieh reduces to P when L'~0, Jtf'-O.It thns appears again that at the two points where LM meets
ail thé snr&ceshave the eametangent plane.There are two other points whosepolars with regard te all
the quadneaare the same, whichwill be verticee of coneocon-
taining both the corvéeof section. It is easy to aee geome-
tncaUy that these two points lie on the polar of thé Une LM
with regard to the sor&ce~8(that Mto say, on thé intersectionof the commontangent planes at thé pointawhere LM meets
jS), anA that thèse points are the foci of the involutiondeter-
mined by the paire of points where that polar meets jS andwhere it meeta Z and M.
1S4. If two Mï&cea each interttect a third in thé same
plane curveand in two other plane curvea they will abo inter-
sect each other again in a plane curve whoeeplane passesthrough the line of intersectionof thé two latter planes.
For evidentlytwo snr&ces<S'J~M, j8+Z~ have for their
mutual intemeedontwoplane sectionsmadeby 2/, Jtf–2?.
186.8Mlatquadn<Nbet<mgtothedaasnowunderd!8-cassion. Two quadricaare similar and simi1arlyplaced whenthé termsof the second degree are the same in both(see
CoM«M,p. 801). Their eqna~oaathen are of the form ~0,<8'+&&'=0. Weseethenthattwos~qnadriosmtersectingeneral in oneplane carve, thé other planeof intemectionbeingat infinity. If there hothree similarquadmct,their three nnite
plamesofintersection paasthroughthe same right line.
Sphères are aUa!nu!arqaadncs, and therefore are to beconsideredas having a commonsectionat iaBnity,whichsectionwill of coursebe an imaginarycircle.
A plane seeëon of a quadrie will be a cirde if it passesthrough thétwo pointain whiehits plane meets this imaginarycirde at infinity. We may see thus immediatelyof how manysolutionsthe problemof nndingthe ciroularsectionsof a quadricis sasoepdHe. For the sectionof thé quadrioby the plane at
FOCt. 9&
H8
inSnitymeetathe eeetionof a sphere by the Mmephae in four
pointa whiehcanbejoinedby six right lines,the ptameopasaingthrough any one ûf whichmeet the quadric in a oircle. Thesmright linesmay be dividedinto three paire,each pair inter-
eec~Ngin one of the three pomte whoae polars are the samewith respect to the sectionof the quadric and ef the sphère.And it M eMy to xee that these three points determine thedirectionof the axes of the qnadnc.
A Mr&oeof revoIntMmN one wMchhas doaMecontactwitha
sphereat infinity. Foraaeqaationofthe&!m<c'+~'+<M'=& bcanbe writtenin the form
and the latter part representstwo ptanes. It M eaey to Me
then whyin thiscasethere is but one directicnof real circalar
section~determinedby the une joining thé points of contactof the McttNtBat inftnityof a qtheMand of the qaadrio.
FOCÏ.
136. When Mpresentaa sphère, the équation of the
quadrichaving dcaNe contact with tt, ~'=ZAf expremes asat CbM«M,p. 216,that the squareof the tangent from any pointon the quadrioto the sphère Min a constant ratio to the rect-
angle itnder the distancesof the same point from two mxed
ptames. The planee Z and Jtf m evidently parallel to the
planes of oa'calar sectionof the qttadnc since they are planesof its intersectionwith a sphère. We hâve seen(<~M!«~,p. 217)that the &cas ofa conicmaybe consideredas an !n&t!te!ysma!!
cMe having doaMecontactwith the conic, dte chord of con-
tact being the directnx. In like manner we may deSne a
&cnaof a qaadnc as an hoMtdy smaU sphèrehaving double
contactwith thé quadric,the chordof contact being then thé
correspondmgdirectrix. That !a to say, the point o~'y is a
focneif the equation of the quadrio can be expremedin thé
formvs 1 Ma i ve n
where~is&epmdoctof thé eqa&tioMoftwo planes. We
mnstdisomaeepMatetyhoweTe)'thé two cases, where these
100 FOCt.
planesare real amdwheMthey are imaginary. In the one casethe equation!s of thé form <8'a'ZJt~in the other N-Z'+Jtf*.In the nrat case the directrix (the UneLM) is paraNelto that
axis of the surfacethrough whichreal planesof dreolar section
can be drawn. Thus, for example, if the aurfaoehe an ettip-MH) the line ZJ)f must be parallel to the mean axm. In thésecondcase the Une LM must be parallel to one of the otheraxes.
Lt either caae the sectionof the quadric by a plane througha fbcnaand the correspondingdirectrix will he a conie havingthé eame point and Hne for focus and du'ectnx. For if wetake the axes <cand y in any plane through LM and thenmake <=(), the equation reduceato (a!–f[)*+(y-~)'==~ or
ebe =<P+m*where Mare thé sectionsof Z, Mby the plane<0. But if this plane paas through LM, theaesectionscoin-
aide, and the equation redncea itaelf to (!B-a)*+(y–/3)*<=?',whichreprésentaa comchaving e<j8for&)cusand 1fordirectrix.
This M only thé algebraioal statement of the tact that the
section in question Is toucbed by the infiaitely amall circlewhich is the section of t8, heing the chord of contact.
187. Let us now examine vhethor a given quadnc necea-
sarily haa a &cns and whether it bas more than one; that isto say, whether the equation of a given quadric can be ex-
pressed in the form ~==Z*j:Jf*, where S Is a point-aphere.Now if the co-ordinate planes <cand y were any two planesmutually at right angles pasaing through ZJ~ the quantityZ*± ~f*would.be expressedin the form<KC'+ S&ey+<!y*,which
by moving round thèse co-ordinate planes could be made totake the fom ~a~i~ And if now the originwere movedto any point in the plane through thé locus perpendiculartothe directnx, the equation ~'=L'i~f' would take thé form
where a, ~8are thé x <mAy of the &)caB,'y,8 thoseof thé foot ofthe directrix, and where, when -4 and B have oppositeaigma,the planes of contact of the focus with the quadrie are real,while they are imaginary when A and B have the eame sign.Oar eo-OKtm&teplanes have manifestly been so chosen as to
MCt~ Mt
he parallel to thé principal planes of the Mî&ce, and we aovwant to 6nd whether by a proper choice of the constante
a, 'y, 8, B, the form just written can be made ideattcalwith a given equation
Firet, in order that the originmay he the centre, we musthâve <t'=~'y, ~=~S, by the help of whieh eqnat!ona elimi-
nating < 8 the formwritten abovebecomss
Thus it appeam that the aur&cobe!ag given the conataata Jtand B are determined, but that the &cna may lie tmywhe~ond~c~M
whiehacoordinglyis calleda~xM?<!OK&of the surface.Sincewehâve pturposetysaid nothing as to either the signa
or the relative magn!tadesof the quantitiea L, ~tf,N, it followsthat there is a focalconîc in «M~of the three principal planes,and <Jso that this conic is confocalwith the correspondingprincipalsection of the surface; thé conies
being plainly con&caL Any point a'~ on a focal conic beingtaken for jbcaa, the comespoNdIagairectrix is a perpendîcuiarto the plane of the eomo drawn through the point
102 MCÏ.
Thèse vatneamay be imterpretedgeometdcaByby saying that
the foot of the Au'ectnxMthe pole,with rea~tectta the principalsection of the surface, of thé tangent to the focalconic at the
pomt<t'/8*. For this tangent is
which is manî&sdythe polarof'/S' with regard to + ~==1.
Hence, from thé theory of plue confocalcMucs,thé I!ne
joining any &<mato the foot of the coïrespondtngdirectrix isnormal to the focal conic.* The feet of the d!rectnces must
evidently lie on that conicwhich is the locus of the poles ofthe tangents of the ibcatcontewith regard to the correspondingprincipal sectionof the quadric. The equationofthis conicis
M appeara by elmuDatmg<t',~S*from the equationof the focalconic tmdthe eqnat!onsconnectinga', ~6','y',S*. The directriceathemBelveeform a cylinderof which the cornejust written is
the base.
188. Let 118now examine in detail thé dMbrentdassea ofeentm~aaï6M!es,in o'Aerto mve&tig&tethe Batoraof their focalcooicsand to imd to wMchof the two dirent kindsof focithé
pointa on eachbelong. Now it is p!amthat thé equation
will repreaent an ellipse whenNis a1gebraica1lythe least ofthe tbree quantities J&,JM,N; an hyperbola when is the
middie, and will becomeimaginary when N !a the greatest.Ofthe three focal conicatherefbreof a centralquadric, one
is alwaye an ellipse, onea hyperbola,and one imaginary. In
It wu pMYedthat the planejeMag any&MNto thee«n'eq)«ndin~diteetiaxmeetathe ttM&eeina sectionef whiohthatpointis the fooue.It appearsnowthat thi<maybe tttttedasapropertyofanytttaaenormaltoa fbtatcotte.
MCt. MS
thé caMof the eUip~H, for exampte, the eqtMtt!onaof the <o<it~
eïUpaeand <bcathyperbolaaM ramectivety
The cojesponding eqoationsfor the hyperboloidof one sheetare foundby cbanging the sign of < and thosefor the hyper-boloidoftwo sheetsby changingthe sign both of b' and <
Parther, we have seen that foci belong to thé claas whoM
planes of contact are imaginary or are real, aecording M
and B have the same or oppoMtesigna, MMtthat~=–,
TM~_xrt, Now if be the least of thé three, both nume-
M
rators are positive,and thé denominatorsare a!Mpositive inthe CMCof the ellipsoidand hyperboloid of one sheet, but inthe ca~oof the hyperboloidof two sheets one of the denaBd-natom is negative. Hence, the points on the focal ellipse arefoci of the daM whoaeplanes of contact are imaginary in theCMesof thé ellipsoidand of the hyperboloidof one aheet, butof the oppositeclasam the CMeof the hyperboloidoftwo aheets.
Next, let N be the middleof the three quantities; then the twonamerators have oppositesigna, and the denommators havethe eame sign in the case of the ellipsoid,but opposite in thecase of either hyperboloid. Hence the points of the focal
hyperbolabelongto the dMa whose planes of contact are realm the caseof the eHipeoid,and to the oppositedass in the caseof either hyperboloid. It will be observed then that all théfoci of the hyperboloidof one sheet helong to the otass whose
planes of contactare imaginary; but that the focal conics ofthe other twosarfacescontain fooiof oppositekinds, the ellipseof thé ellipsoidand thé hyperbola of the hyperboloid beingthose whose phmes of contact are imaginary. This is equi-valent to what appeared (Art. 186)that foci of the other kindcan only lie in planes perpendicularto that axis of a quadricthroughwhiehreal planesofciïco!ar sectioncan he dmwn.
139. Foeat conieswith real planes of contact intersect the
sar&ce,wMe those of tho othc)' kind do not. In &ct, if the
FOCI.t04
equation of a earfaoecan be thrown into the form <6''=' + Jf,and if the co-ordinateaof any point onthe Mt&eemake ~'=$,
they must aîao make Z. =0, M= 0; that !s to say, the focusmuet lie on the directrix. But in tMacasethe MH&cecould
only be a cone. For taking the origin at the focus,thé equa-tion a!'+~'+~=*Z'+Jtf', where L and jtfeach pam throughthe origin, would contain no terms exceptthose of thé highestdegree in the vanaMeft,and wouldthereforerepresent a cone
(p. 40).The focal corneon the other hand, which mctades foci of
the Srat kind, pâmesthroagh thé umbilica. For if the équa-tion of thé Bar&cecan be thrown into thé form ;S'=Z3f, andthé co-ordmatesof a point on the surfacemake <S''=0,theymuet a!somake eitherL or J<f=0. Bat sincethe sarface passesthrough the intersection of ~8,L; if the point S lies on Z, the
plane L intersectsthe surface in an infaitely Nna!Icirde that
is to say, ia a tangent at an ombilic. From this propertyProfessor Mac CaHaghcalled focal conies of thia latter kind
MMMMMM'focal conics.
The &callimesof the cône, asymptoUcto any hyperboloid,are plainly the asymptotesto the focalhyperbohof the surface.
The foci on the focal lines are ait of the dass whoseplanesof contact are Intaginary; but the vertex ItM!<,heBtdeabeingin two waya a &<:mof this kind, may abo be a &CNSof the
fOCï. 106
otherkM, for the equationof the cone can bo written in anyefthethKefbntM
The directrix, which correspondsto the vertex coBBtdetedasa &cnB,passes through it
The !!nejoining any point on a focal line to the foot of
the corMepondmgdirectrix ia perpenct!cn!arto that focal line.This followsas a particularcaseof what haebeen alreadyprovedfor the focalconicain gênerai, but may a!sobe proved<Hrectiy.The co-ordinatesof the foot of the directrix have been proved
aeenusatMned.ln like manner, Ma part!cnlarcase of Art. 186,the section
of a cône by a plane perpendicular to either of ita focal Unesis a con!cof whichthe point in the focal line is a iocns.
141. ï~ ~XM~~MMtof a <WMare perpendicular to Me CM*-
ct<XM'Mc<«MMo/*<~t~e~oMt~eoMe(seeEx. H, p. 87).For the circular sections of the cône Za!*+<~+~e*0,
are (see Art. 99)
and the aorrespondingfocal tines of thé reciprocal cone"i
9 a! !IZ
are as we have just seen r–M-+M~=' Ot
Mtdthe Unesrepresentedby the latter equation are e~den.tlyperpendicularto the planes representedby the former.
FOCI.i06
The theoremof thia attide !aa particularoaaeof the Mov-
ing more genefat:–?!~ «e~tOMof <!<?Mo~Mo?cotMt <M~p&MeaMpo&!ftW~Mea& Mt<ty~a~<o<ejpetyca-«'&M&t~on<&<t<~&MM~MthecoNMMOMcet~c. For let the planemeet an edge of one cone in a point and the perpendioulartangent plane to the other in the line (?, let 3f be the foot
of the perpendicularon the plane from the vertex 0, then it
MeœytoMeÛwt&eUMJMfNpM~m~d~to~N~Mdif it meet it m iheo Btnccthé triangle FC~ is nght-MgM,the rectanglejR!f.~fN is equalto thé constant0F*. The eurvetherefore which is the locus of thé pointeP is the same as that
got by letting fall from M perpendictJarson the tangentsand taking on eaoh perpendiculara portion inveme!yas its
length. When thereforethe sectionof oneconeis a cirde, thatoftheotherwiUbeacamcofwhichJtfisa&K!a& Weahaltdiscuss with more detail the propertiesof comeawhenwe treatof apheïo-conics.
142. The investigation of the fociof the other species of
quadrics proceedain like manner. Thus for thé paraboloids
motnded in thé equation y + ~c'2~. ThN equation can be
written ineither of the &rms
It thus appeaMthat a paraboloidhaa two focalparàbobB,i~Mhn~Mm~~M~a~~eMhM~M~~Mh&eMm~-spondingprincipal section. The jbeaebelongato one or other<f the two kinds aheady dMCNSsed,accoKtmgto thé sign of
y ~f
the&actMn–y–. In the cMe ofthee!!tpt!op<uKtMoM
MCt. 107
thme&M'ewherebothj&and3fa)'epeMt!ve,!f~bethegteater, then the foci in the plane <caare of the daas whoee
planes of contact are imaginary~whilethoM in thé ptano~eaM of the oppositedose. But aince if we change the sign
eltherofZorofJMtteqa<mti<y–-y–renMMnaposMve,weseethat <tKthe foci of the hyperboMoparaboloidbelong to theformer daas, a propertywe have aheady oeemto be trae of thé
hyperboloidof onesheetIt remainatrae that the line joining any &MaMto the foot
of the correspondingdirectrixis monnaï to the &cal cnrve, andthat the foot cf the direetrix M the pote with regard to the
principal section of the tangent to the focal comc. The feetof the directriceslie on <tpambola and the directrices them"selvesgenerate a pM'aboHccylinder.
To completethe diacnBmonit romaine to notice the foci ofthe dMerent kmds of cyHndem,but It Mfound without the
eEghtestdUScaltythat whentho base of the oylinder is anellipse or hyperbola there are two focal Unes; namely, linesdrawn through the foci of the base parallel to the generatorsof the cylinder, while, if the base of thé cyMnderis a parabola,there ia one focal line paasingin like manner throngh the &cneof thebase.
148. The gcometncal interpretationof the equation ~~Z~fhaebeen aiready given. We team from it thiaproperty of fociwhoeeplameaof contact are real, that the <~«tMof <&ed'M(oMee
of any point on 0 <p«MM)J~WWtMM&a J<!MMM in a <!MM<M!<
M<M<0 <itejMW&tC<~~epetyeK~MM&NW~t~M~MM ~~MM<M~e~M~w~~MN~w~~M~A~MM~M~wM~
<M~o<Ma~jpaa~M ? the planea of circular <ee<<OM.Thé cotre-
aponding property of foci of the other kind, which ta less
ohvîoua,wu diecoverodby Profeseor MacOnHagh. It is, that
<~aftS&MMeû/'«~jpOM!<on <~<a'M<C~M «M&OJ~CtM&a <!MM<tM<M)<&<0ita dSM(aMCej~~M<XM~p<M<)~<~M~C<H~<i~&!«ef<S<<<Mcete~ <tM«M<Mt?~<!MtBM~e&~ ~'tSejp&M~~<!&~M&M-<eC<MM.
Sappose;in&ct, wc try to express thé distanceof thé point
K)CT.108
a~V from a d!rectnx parallel to the axis of < and pMsmgthrongh the point who<e<eand y are 'y, the distancebeingmea<uredparallel to a d!reettveplane <fMa:. Then a parallelplane through a!'y.c',viz. <e''='M(ic-a)'), meets the directrixin a point whose a; and 11 of c<mmeMe -y,8, while ita o ta
given by the equation <-<'=<? (-y-a)'). The square of thediatanoerequired is therefore
where .A and B are both positive and is sapposedgreaterthan .B, the nght-hand side denotes B times the square of thedistance of the point on the quadric from the directrix, thedistance heing measnred parallel to the plane a~NM; where
M~'es–By pat~ag in the values of and B, givenB
in Art. 187, it may be eeen that thia ia a plane of clrcular
section, but it ia evident geometrtcaHythat this mnst be the
case. For consider the section of the qnadric by any planeparaUelto the directive plane, and amceevidentlythe distancesof every point in anch a section are measnredfrom the same
point on the directrix, the distance thereforeof every point in
the section from tMs fixed point !s in a constant ratio to Itsdistance from the focus. But when the distancesof a variaMe
point from two nxed points have to each other a constant
ratio, the locusis a aphere. The sectiontherefore is the inter-
section of a plane and a sphère that is, a circle.
An exception occars when the distance<rom the focus Is
to be equal to the distance from the directrix. Sincethe locus
of a point equidistant from two nxed points is a plane, it
appears as before, that in this case the eectionsparallel to the
directive plane are right lines. By re&rnng' to the previousarticles it will be seen (soeArt. 142) that the ratio we are
eonmdering !s one of e<toaHty(~~1) only in the CMCof the
hyperhoticparaboloid,a sar&cewhich the directiveplane could
not meet in circular sections, seeing that it bas not got any.Profesaor MacCnllaga caUsthe ratio of the focal distance to
Mf!ï. !(?
that &omthe directrix, the modulus of the snr&ce, and the &ct
having imaginaryplanesof contact he cath modular foc!
144. It wu observed (Art. 133) that au quadrica of thoform tS–Z~f are envelopedby two cones,and when 8 repre-aentfta sphère,these conesmust be of revolutionas every cône
envelopinga sphère must be. Further, when<S' reduces to a
point-sphere,these cones coincide in a singleone, having that
point for its vertex; and we may t~ere&re in<~rthat the cone
envelopinga quadricand having any &)cusfor ita vertex Moneof revolution.
This theorembeing of importance we give a direct a!gc-braical proofof it. First, it will be observedthat any equa-tion of the form a!'+~s*=(<B+&y+ee)* represents a rightcône. For if the axes be tranaformed, remaining rectangclar,but so that the plane denotedby <M)-t-<y+o.:may become oneof the co-ordinateplanes, the equation of thé cone will become
JC*+y+~*='~ï' wHch denotes a coue of revolution smce
thé coeincientsof Y*and are equal.But nowif we fbnn, by the mte of Art. 74, the conewhose
vertex is the origin and circamaeribmg iB'+y'-t.Z'-Jtf,where
ïntheyeM!8aaPM&MorMMCaU~hpaMahedthMmodulMmethcdof genettttion of quadrica. In 1842pubUshed the supplementary property
poMeMedby the non-modulu toc!. Not long a&er M. Amyot indepeu-dently noticed the aame property, but owmg to hh not being acquaintedwith fto&MM MMCuttagh'e method of generation M. Amyot Med toobtain the complete theory of the fooi. FtofeMor MacCaHagh bas pub-tiehed a detaSed MCMmtof the focal propertim of quadrics, wMch will befound in the J!~o<o~~ < «e Royal Irish ~M<&My,Vol. n. p. 446.
]M)t.Townaend aho bas pnMMted a valuable paper (ChtttMa~ and JDaMMJM<!«eM<!<&e<J<M~<~<\ol. ni., pp. t, 97, 148) in whieh the properties of
foei, Mmideted M the limite of apheres havia~ double contact with a
quadrie, are very MIy inveetigated.
METKOM Of ABMMED NOTATÎONT.110
whieh we have seen repreeenta a right cone.
A few additional properties of foci eaaily dednced from the
principles laid down are kft as an exetciee to the reader.
Ex. 1. The polar of any direetnx t< the tmgent to the foctd coaic atthe CMKtpondhtg &MM.
Ex, 2. The polar plane of any point on a dheettix i< pMpendteuhr te
theMMJMaiagthatpotnttotheMn'Mpoïtttin~&OtM.
Ex. 3. If !me be drawn throngh a &Mdpoint 0 emtt!ngany directrixof a quadric, and meeting the quadrie in the pointa B, then ifJPbethe
corresponding (be<M,tan~FO.tm~JM) i< constant Thh is proved M
the CMtetpondingtheorem for plane emies. CbftfM,p. Ml.
Et. 4. This remaiM tme if the point 0 move on any other quadriohaving the Mme &)ea*.directrix and planes of oiranlar aectien.
Ex. 6. If two auch quadrica be eut by any KM paasing thmgh thé
eoannoa di)'eetd<, the angles mbtended at the &Mo<tby the intercepteare eq)ML
Ex. 6. IfaHMthMagh~diKo~t<mAoMoftheqMdm!t,thech<)tdiatefeepted on the other Motendt a e«nst<mtangle at the ibeat.*
146. Having now considered the most renMu'kaUe cam of
quadrics mduded in the eqnation j6'-ZJM,f let as pase on to
the eqnation jS'-2<*==0, whieh denotes a ani&ce touching <8'all
along the section of by the plane L. It M eamiy ahewn
from geometncat conaideranons, as at Art. J98, that tvo quadricscannot touch in three points withont thus touching aU along a
plane curve. The equation of the tangent cane to a surface,
given p. 48, M a particular case of this equation jS's L*. AIso
two concentnc <md aunHar quadrica are to be regarded aa
In this <eet!on an aeconnt hu been given of the relatiom wUeh eaoh
fbeM of a quadrio considered sepaKttetyheMt to the Mr&oe. In the next
ohapter we ahall give an eeconnt of the ptopettiM of those eonies whioh
Me the MMmMttge of foci, and of eentbtat Mtiheee. Thete ptopet~eowere aMt etadied in dettdt by M. ChMkt and by PM&MOfMaeCoitaghwho about the same time independentty arrived at the ptineijMl of them.
M. Chatte~ reaulte will be found in the notes to bis ~p«'ft< J5!«t<ett<pM,
paMiehed m t8S7.
t Thé eaMwhere 'S'breatMnp into two phnet bu been dhtMMd p. 78.
M)BntOD8 ? ABBÏD&BB NOTATION. 111
envelopingeach other, thé plane of contact being at in&tity.Any plane obvioady cata the sor&ces and <8'-L' in twoeonieshavingdoublecontactwith each other, and if the sectionof one reduoeto a pomt-<arck,that point mmatplainly be the&cuaof the other. Renee <oA«!one gtMdWceNee&p<'<<tKo~<rMe &M~M<plane <t<the MMMtCof one cute ~e other in a<!M!Mo/' <eAM& «MMScM <~ j~cMs/ or if one sot&ee be a
sphèreevery tangentplane to the sphere meetethé other sar&oein a Mc~onof whichthe pointof contact is the &caa.
Or theae things may be seen by taking the origin at the<mtbiHoand the tangentplane the plane of xy, whenonmaking<=0, the quantity ~-J!<*reduces to a~+y* and denotes aconicof whichthé originia the focus,and?thé dtKetnx.
Twog~Md~MenMbpar theeame<~M~M~'MC<«M&other
t<tjp&M!ecMn)M.ObviouelyS–JD', <S–3f*, have the planesZ L +Jtf for their planesof intersection.
146. TheeqnatMn~'+~+~+~,whereZ,jM;Pfepresent planes, denotesa quadnc sach that any one of thèsefour planes Mpolar of thé intersection of the other three.For ajD'-t-&Jtf+oy denotes a cone having the point .Mf~for its vertex, and the equationof the qnadrMshewathat this
cône touchesthé qnadric,P being thé plane of contact. Thefour planea form what 1 sha!l cat! a ee~<<~a<e tetrahedronwith regard to thé snrtaoe. It bas been proved (Art. t86)that given two quadricathere are aiways four planes whose
potes with regard to both are the same. If these be takenfor thé planes L, JM, J~ the equations of both can betransformedto the &nnB
It might aïaohave been aeen, a jM~<~ that this is a formto whiohit muetbe pomibleto brmg the eystemof equations
oftwoqmMhics. ForZ,JM,J~,jP!nvolvetmpMttythreo<xmi~tseach;Mdthe~uat!oDs~ttem&bw<mvolveex-plicitlythree independentconstantseach. The systemthere&re
md~Me~~MnMm~M~Mt~K&MS~M~~ygmM~to expressthe eqatt~omof any two qttMhics.
METHOM OP ABNïMED NOTATtON.112
147. 2b~M <&<«MMHttMt<X<t<onepKKfrMa~MtHjMMS<&<W~ttXeoe~MMof a<c~<!<~<~<t<e<e<M<«&wt?<? f~<tr<?<oanother.
If a), y, te denote the &ees of such a tetrahedron, then
the equation of the one quadric expreasedin terme of theseasaameathe form <M~+~*+o<+<?<e'c'0,while in the equationof the other, the coeSctenteof a~, y', < td' vaniah. Now if
we formthe discriminantof ~+XF, whiehwe shaHwrite
A+\0~<t'+\'@'+~A'-0,it willbe seen that if all the terma in Uexcept<ï,&,c, d vanish,then 0 becomeaa'M+ &'<!<&t+ o'<M+ (Me,whichvanisheswhen
<t', y, c', d' vanish, and since the coefficientsA, 0, &c. are
invariants,@ will be c'O, no matter how thé axes are trana-
fbrmed, if Y pass through thé ver6ces of a self-conjugatetetrahedron with regard to n
When U redaceBto aa!'+~-t-<M*+J<e*)the quantity 0' ta
<tA*but
~=0is the conditionthat the plane<cehould touch the
surface F. Rence @'=0 m the condition that the faces of &
se!~c<Mt)Og&totetrahedron with regard to P' ahoaMtouch the
snr<~MeF, as weUaa thé conditionthat the vertices of a self
conjugate tetrahedron with regard to shoold lie on thé
surface C. If, therefore, one of theaethings be thé case, theother must atao. <0 willbe fnlûlledif the edgea of &ae!f.
conjugatetetrahedron with regard to either all touchthe other.
Ex. t. If a apherebeeircMBtonbedabouta Mtf-coojug~tetetrahedron,the!engthof thetangentto it&<Hnthecentreof thequadrieis constant.
For whenF Ma spherewhoaecentreMa, /9t?and radiusr, and
KETHOM0FABRIDGEDNOTATMN. m
t
Théequation9 0 MoMaethe theoiamentmextted.ThetNreqxmdingthcoremforecn!Misdueta M.Faure.
Ex.2. If a hypefhoMdbe Mchthat t,. 0, thenthe oentre
of a apheteimetibedin a Mt~eo~te tetrahedrontiMon thé Mîface.
Et. 8. The toe<Mof the centreof a ephereoKnmMfibmga Mt~
co~ugttetettahedronwithMgMdto a pttmtoMdis a plane.
148. The verticea of two se!f<onjngate tetrahedra with
regard to a qnadrîc, form a aystem of eight points,auch that
every quadric through aeven will pMa through thé eighth.Hesse, C~e~ t. XX.p. 297.
For, if a:, e, w, F, W be thé &cea of the two
tetrahedNt, the quadric caa be MpieaMd in either of thefbrms
a-, y, &o. Mng sapp<medto contain constantmnltipliere im-
plicitly. Now if any quadric given by the general eqn&ëonin a:, y, <,<cwere transformedto a faction of Y,Z, W,we find,&omthe invarianceof the fonction@,
snd conseqnently,if sevenof theae qusnëtiea i~nushso muetthe eighth. In like maïmerany quadriowhich touches sevenfaces will touch the eighth.
149. Thelinee~&tM~ <~ Mf<M<Mof any <e()~e<&wt? the
cMTMpMMK~oerttCMof ita ~)o!<!f<e<a~a~M:M«~f~o~J ? <t
~tHKMcMM!~to the «MM~<<~Mû/MM<Ofe of a ~~pM'&jMof one <~ and Me Mt<a'MC(«MM0/' C<MVMpOa<~M~J<!MMof the<wo<e<M<a~ajMSMMtheMmejM~pe~.
The Kso!t of aubstttatmg the co-ordmatesof any point 1~m the polar of another point 8, M the Mme aa that of anb-
s6t[[t!agthe co-ordinateaof 2 in the polarof 1. Let this remitbe called [1,8]. Let the polar of Ihe caHedJ~. Then it M
esay to Béethat the Une joiningthe point 1, to the intersection
of~F.b
METHOM OP ABBtMED KOTATÏON.114
For this denotesa right line pasaingthroughthe intersectionof J~, and whoseequationis M~anedby the co-ordinatesof 1. The notation will be more compactif we cati the four
polar planes x, y, <,<c,and denote the quantities [1,8], [1,8],[1)4] by M,M,~ that Mto eay, by the eameletters by whïohwehave expreesedthe eoeS)<aentaof a~ aN,<Mpin the generalequationof a quadric. Then the equations of the four lineawe are considenng are
Now thé conditionthat anyline
<!ic+~+c<t+<~e=0, <t'«!+~+o'e+<fM='0
ahouldintenect thé Ërst, Is, by dunmattBg<Bbetween the lasttwo eqMtKHts,jbnnd to be
)t(<
MKfHOMOf ABNMEONOTÂttON. 116
t2
Tho eqnation of the hyperboloid itself is found ty themethotb of (p. 78) in the form
1SO. The secondpart of the theorem is only the polarrectproctj of the &mt,but, as an ex~MMe,we give a separateproofof!t.
Let [1, l], [1,2], &c.have the same mgni6catlonM before,viz. the resuit of snbstitating the co-ordinatesof 1, in the polamof 1, 2, &e. Fona thé determinant
METHODS OP ABBtDGEBNOTATION.1M
and, as before, the theorem is provedby thé f&ctthat thesecondMomswhemadded vamishidenticaHy. The equation ofthe
hyperboloidis fband to be
As a particular case of thesetheoremsthe lines joining eachvertex of a cireumecribingtetrahedron to thé point of contactof theopposite face are generatoreof theeamehyperboloid.
151. Pascal's theorem for conicsmay be stated as follows:Thésides of any triangle intersecta conic in six points lying
inpairs on three Uneswhich intersecteach the opposite s!deof
the triangle in three points lyingin oneright line." M. Chaslesbas stated the following as the analogous theorem for spaceof three dimension: The stdesof a tetrahedron intersect a
quadric in twelve pointa, through whîch can be drawn four
planes, each containing three points lying on edges passing
through the aame angle of the tetrahedron, th~n the lines
of intersection of each such plane with the opposite face of
the tetrahedron, are generators cf the same Systemof a certain
hyperboloid."Let the faces of the tetrahedron boa?,y, w, and the quadric
whoseinte~ectMM with thé planesa!)y, <p,respecttvdy are
a systemof lines proved in the lut ardde to be generstoMof
the same hyperboloid.
METHODS0F ABNMB5NOTATION. 117
152. As a ~trther illaetration of thé use of thé invanantt,in Sadîng thé conditionswhich express the permanent relationsof two quadn~ to eachother, we mvesttgatethe conditionthat
two quadrics ehaUbe auch that a tetrahedron may have two
paila of oppomteedges on the eoïfaceof one whUeita four
facestouch thé other.* The one quadricthen can be made toaasnme the formJ'~Mp+ I~jj! ==0. If the four planesiB, e, «' °
touch a quadric tte equationwill be foundto be of thé form
~-i-+e'+<0'+a~(<B!P+~)+2M(yM+<~)+aM(<!M~a~)=0,
where l+2&Ma=f+M'+M', amï if ~M,M be each iesa than
unity, we may write for them -coe~, -coa~ -co8<7, where
J?, are the angles of a plane triangle. It will be round
then that
163. 2~e~<<~M~<~cM'emtMCMM~<!<e<Mi~e<<5*oK.
Let the four facesbe a, j3,y, 8. Let the fourperpettd!cuhraon each face from the opposite vertex be p~ Nowthe equation of the c!ro!ec!rcumscnbmgany inangte abc mayho written in the form
This appeara to be the problem whieh corresponds to thé planeproblem of 6nd!ng the condition that a triangle shatt te hMcnbed in one
conieand eireunMO'ibedabout another.
MMHOMO? ABMMtEBNOTATtON.118
where a, j~, &c., denote petp<md!cn!atson the tmdeaof the
triM~e, the lengtha of wMeh are (~o),&p. But it ia evidentthat for any point in the &oe S, thé ratio a:~ ia the samewhether a and p denote perpendîcaittraon thé plane a, or
perpeadIcalMson the MMa8. We are thueled to the equationre~tmed~viz.
For this is a quadrio whose intersectionwith each of thé
four &CMis the circle cu'emaMïibmgthe triangle of whichth&t&cecoBBMta.
It w!Rbe &and) that when the equationof the sphère in
Wtitten in the above &ttm, the coe&dent of y*,< ia 1.
Hence the square of the distance betweenthe centres of theuMcrihedand<arcamscnbmgsphereeN
1M. Fromthe preceding equationwe can dedace the eom-ditions that the general equation ahould represent a sphere.For the equation of any other sphère can only differ fromthe precedingby terma of the ûmt degree, whieh will be of
thé&rm(&+~8+CY+~8}~+~+~+-
thé secondP p y jp
&ctor denotingtho plane at innnity. If then we add to the
équation of thé lut article the ptodact of theae two factors,
identify with the general equation of the aeoonddegree, andeUmmatethe mAetenoinateconstants,thé resal~Bgconditionsare foundtobe
MNHOD80F ABMDOEHNOTAMOM. 119
1M. Given two quadrics U and F there ara two other
principal covMMntquadnee i& terms of whieh toge&er withU and F and with thé inTarMmte,aUother covariantquadricscan be expïesaett. We sha!! choose M those two covariants,jS'the locattof the poteawith rMpeet to U of <tUthe tangentplanes to F, and <8'the loco<of the poles with respect to 7of all the tangent phnes to (seeEx. M, p. 87). Thueif
These quadrics it will be observed,M we!I as !7 and V;have
x, y, «, Mfor the faces of a sot~con)t)g&tetetrahedron. Hence
we cat)solve the problem,given two quadrics !7 and F, to 6nAthe equation which denotes the four planes a:, y, < <owhose
poles with regard to both are the same. For we form the
covarumtB and <8'and then we haveonly to form the Jacobianof the four functions that ia to say, the determinant
whosefour rows are
whenwehavea functiondenotingthe four planes in qnestton.
156. The conditionthat eM+~+'ye+Stc ahoatdtouch U
is a contravariant of the third order in the coem!c!ents.If we
mbBtttntefor each coeJË~enta, <t+\a', &c. we ahan get tho
conditionthat fM+~ + 'ye+&cshall touchthe surface0'+ F, a
conditionwhichwillbe ofthe formo- +X-r+X'T'+ \'</ =0. The
6uMttoaa<f,<r',ï, T'eachcontaina, /3, &o.in the seconddegree,and the coe~deats of U and F in the third degree. In termsof these fonctionswe canexpressthe conditionsthat the sectionsof FandFbytheptaneaai+~y~yo-t-~MBhatihaveanype~manent re!a~on to each other, snch as cm be expreMedinterma of the eoeSMentsof thé discriminant of P+\Fwhcn
METHOD80F ABMD8ED NOTATION.120
U and F are two plane entrea. For instance,the condition
that az+ ~y+'ye + OM'shoaldmeet thé two sm&cesin sections
whicbtouch,!s got by formingthe discriminantwith respect to
of <r+\T+X'T'+X*<='0; or, in otherwords,this expressesthe condition that the plane <M+~+'ye+&o shoald pass
throngha tangent Uneof the curveof intersectionof U and K
This conditionwill be of the eighth order in a, ~9,y, 8, MM~
of the sixth order in the coefficientsof eachof the ear&tcee.
157. The conditionthat a!C+~+'ye+&o shonidtouch U,
may abo he regarded aa the equationof the surface teciprocatto U withregard to a!* +~+ +<o', (Bée<XM«~,p. 268). And
in likemannera-+ Xr+ X*T*+ XV !s thé equationof the sar&ce
reciprooalto !7+X~ Since if X variée, !7+~F denotes a
series of quadrica pasaing throngh a commoneurve, the re-
Ctprocatsystem touches a commondevelopable,thé equationof
whichisfoundbyformingthe discriminantof or +XT+X'T'+X'<r'with respectto The equationthereforeof the developable
reciprocal to the curve of intersectionof U and F is, as has
been noticed in the last article, of the eighth degree m thenew vttnaMes,and of the sixth degreo in the coenidents of
each of the surfaces.
By the same method we can form the eqnation of the
developablewhich touches both U and K For têt <7and Fbe the aar&ccsreciprocai to U and then the reciprocalof
!7+XF wffibe a snr&ce inscribedin the same developableas
U and P, and the discriminantwith respectto Xof its equationwill be the equationof the requireddevelopable.
1M. By the help of thé canonicalform eK~+ + ce' +<~M*we canreadily express thé circumscribingdevekpaMein termsof U, F and the two covariantsjS' and;8'. Let Uthe reciprocalof U be ~~+~3*+C/-t~ then ~=&:d, &c., and thé
reciprocal of P' will be BC'P.c*+&o.,that is to say, A'K
Again, the coenicientof Ms (~CZ)'+ C'DF+2)J?C')a!*+&c.,whichM=*A<M'(&'c'J+<<y&+<f&'c)a~+&c.,while thé quantitymultipliedby A is
METBOM 0F ABMDSED NOTATMN. 121
The developablethen iethe discriminantof
The discriminantbeing cleared of the irrelevant &ctor A'A"
the reault remaineof the ten& degree in the coen!c!entsofeach
equation. jS' and evidently pM8 tbrough the ccrves ofcontactof the developablewith U and f~ whilethe develop-able meeta U again in the CHrveof intetaectioBof U with
(Op-- ~)'+4A~, (MeArt 160, t~a).
1S&. 2b ~tj MK~iMMt~<t< a ~tMM?« <(M JMMthroughthe Ct~Mof intersectionof <<PO~MOMMMU and K
Supposethat wehavefoundby Art 76 the conditionp'=0,that the Urneahouldtouch 0, and that we aubstttntem it for
eachcoefSc!enta,o +X<the cond!t!onbecomeap +Xer +\'p' = 0;and if the line have any arbitrary position,we can by solvingthis qaadrat!c for determine two snt&cee passing throughthe curve of intersection CT~tmd touching thé given liae.
But if the l!ne itself pass thmugh UV, then it !Bettey to see
that both theBesm&ceBmust coincide,for that thé !me cannotin general be touchedby a surface of the system anywherebnt in the point where it meets CT~ The condition there&<re
whieh we are seeking is o~!=4pp'. It is of the eecondorderin thé coefficientsof each of the mr&ces, and of the fourthin the coeScientsofeachof the planes determiningtheright Une.
The condition<r=0 wiUbe fulfilled if the right lino is cnt
hannomca!ly by the two surfaces. In the caae where the
quadrics are <M!)-&+<+~to*, a'a~+~'+c'~+<??*, andthe right line ia oae+~+~+Ste, a'a!+~'y+'/z+y<o, the
quantity p M(eeeArt. 76) S<t&('y<S'-<y'S)')by which notationwe mean to express the sum of the six terms of like formsuchascd (aj8' <t'~8)'&c. Then o-will be S(aS'+ ~t')~ ~)',andcr'–4~p'is
160. ïbj~tJ (&<e~xaftOMof the <!M)e!~<!&!egeneratedby the
<am~K<?<StMa~'<Xecurvecommon<o<7«K<!
If weconsiderany point on any tangent to thie curve, the
MBTHOM OP ANHDaBD NMATtON.122
polar plane «f tMs point with regam to either U or V paMM
evidentlythrough the point of contact of the tangent on wh!ch
it lies. The iatemectïontherere of thé two polar planesmeets
the cnrve F. We findthereforethe equationof the develop.able required by mbttttatmg in the condîtionof the last atUda
developablewill then be of the eighth degree tn thé variables
and of the sixth in the coefMentaof each sar&ce. When
we use the canonical form of the quadrics, it then easHy
appearathat the rMatt M
When we make in the above equation tp=0 we obtain a
par&ct square, hence each of the four plamesœ,y, e, <omeetathe developable in plane cnrvea of the tourth degree whiehare donble lines on thé enf&ce.* ThM is, a priori evident,
mnMtti8pIam6~m<heaymmet~ofth~ngaTe~th&tthMughany point in one of thèse four planes thfongh whidt one
tangent Imeof the cNrvePFpMseN~asecond tangent cana!sobe drawn.
By the help of the canonical form the prenons result can
heexpïesstidintenms of the covariant quadrics when the
developableis found to be
Thé curve !7FM mam&stly a double Enef on the locus repre-
See OM<tn%eand JOMM~JM<!<~NM<MJ~~tMM~VoL m. p. 1?!,wheM,though only the geomotricalproofis given,1 h&darrived at theKNtttby Mtaat 6M'nm<ionof the equationef the developable. See<M<<.
VoLn.p.OO. TheeqMti<~w~alMWMkedentbyMt'.Cfty!ey,t6H.VoLv.pp. M,66.
t It Mproved,Mat jH~i6«'f&t<M<~<fM%p. ?, (Meabo p. 76 of this~nme)t that whea the equationof a Mt&eeis P~ + PT~ + y*x, then
MCÏPROCAL 8MM.CE8. M9
eemtedby thïa equation,aa we otherwiseknow it to be, andthe locus tneeta C' agtUnin the line of the eighth order de.termined by the intenection of C~w!th
This is precMeîythe same equation M that found m Art. M8,and one can aeegeometrically that the line of the eighth order19in &ct the eight tangents to !7F at the points where !7Fmeets 8.
BMIPROCALSURFACES.
161. Although we have made &~e use aheady in this
chapter of the methodof reciprocation, we wlstt now to enterinto a little more detaUon the theoryof reciprocalsnï&cea.
To the section of a surface by any plane coErespondathe
tangent cone which can be drawn to the reciprocal surface
through the correaponding point; and in parëontar to thesection of the one by the phme at infinity con'espomtathe
tangent cone whieh can be drawn to the other thiangh the
ongm. Hence thé asymptottc cone of the one enrfaceM M-
<tpM)co!to the tangent cone which can be drawn to the otherfrom the origin; in the sense that each edge of the one côneis perpendicular to a tangent planeof the other.
Hence abo when the origin is Mt<)5<M(<a quadric, that is to
say, M tmchthat real tangente can bo drawn from it to the
surface, the reciprocal is a hyperboloid; when it is inaide itis an eUipaold when the origin ta on the surface, the tangentplane at infinity touches the reciprocalsurface, that !a to say,the reciprocalIs a paraboloid.
!7P'ieadoublelineoutheMt&ee,thetwotmgent~at anypointofitbeSnggivenbytheequationt~y t <w~+<~x',wheret4e are the tattgentplaneaat that pointto U <nd Mtd ta theKodtof Mbat!tutingin théce*Md!MtMefthatpoint.Apptyi'~thiate&eaboveeqaatMnttNimmediatetyt'oundthatthetwotangent*au givenbytheeq<MtMn(N'<t-~y-'O,wheM}n~,<fthec<~otdiMtettofthepamtaKMppoMdtobetmh«itnted.Thusthetwotangentptanesat e~y pointonthedoublecurvecoincide,andthecm~eiaaceoit~inglycalleda enapidalcaMeonthem]t&ee.
MCmM)CAÏ<8UMACE6.124
The reciproca!of a nfM ear&œ (that !s to say, of a sm&ee
generated by the motion of a right Une)is a roled Mï&ce.For to a right line correspondsa right Une,and to the surface
generatedby the motion of one right !me will corrMpondthe
a<N~acegeneratedby the motionof the reciproctJ Une.* Henceto a hyperboloidof one aheet alw&yscorrespondsa hyperboloidof one sheet <!a!eMthe origin be onthe snrfMewhen the reci-
procal is a hyperboBoparaboMA.It waaproved (Art. 144)that thé tangent conewhoaevertex
!as &)caa!s «Mof revolution,hence the yee~Mwa!of a ~«a<Mt:<Ct<&respecttoa pointonajTMO~COMtC&a surfaceC/'M~M~MK.
162. The equationof the reciprocalof a qutMtncgiven bythe general equationis given in Art. 75. The reciprocalof a
central BHrEMewith regard to any point may alao be foundM &t OMttes,Art. &20. For the length of the perpendieularfrom any point on the tangent plane is (seeArt. 85)
163. The redproca.1 of a sphère with regard to any point
!a a snf&ce of revolution round the transverse axis. This maybe proved as M CoMM, p. M9. It M easily proved that if we
have any two points A aa(t J?, the distances of thèse two pointa
from the origin are in the same ratio as the perpendicular from
Mr. Cayley bas remarkedthat <~ <i!t~'Mof any rukd w~ee &equalto <<<<« of itt twe~MMMtThe degree of thé reciprocal is equal to
the number of tangent planea whieh «m be d~wn through an N'HtMtyright Une. Now {t will be f~rmaUyproved heteaftet, but is M~MenttjfMide&ttn it~eK, that the tangent plane at any point on a ruted aaï&ce
containathe ~enett~n~ line whieh p«M8 tbrough that point. The degreaof thé reciprocal ta therefore equal to the number of genemting lineswhich
meetan <u'bittM'yright Une. But this Mexaetty the ttumbe)*of points in
whiehthe arbitrary line meets the emc&tce,aiMe every point in a (~netatm(;MneN a point on thé surface.
&EHHM)CAL8UMACE8. 125
each on the plane eorrespondmgto the other (C!MM<M,Art. 98).Now the ditttanceof the centre of a fised sphere from thé
origin, and the perpendicular from that centre on any tangent
plane to the sphère are both constant. Rence, any point on
the rectprocatsurface is euch that ita distance from the originN in a constant ratio to the perpendicular let fall from it ona nxed plane; namely, the plane corresponmngto the centreof the sphère. And this locus is manifestly a surfaceof re-volution of which the origin is a focus.
By redproc&tingproperties of the sphère we thus get pro-
perttes of enr&cesof revolutionround thé transverse axj& The
M~-hand coinnmcontains properties of the sphere, the right-hand those of the surfacesof revolution.
Ex.1. Anytangentplaneto a The linejoining focuato anysphereia perpendicularto the line pointon thé surfaceK perpendi-joiningits pointof contactto the oularto the plane through&cMcentre. andtheintersoetionwiththedireo-
tdx planeof thé tangentplaneatthepeint.
Ex. 2. Everytangentme to a. The ceae whosevertexfa theapheteu)a rightoone,thetangent &)cmandbaseanyplaneMotionh
planesaumakingequalangleswith a rightcone,whoseajusiathe Jinetheplanoofcontact. joiningthetocuato thepoleof the
planeofMotion.
A particular case of Ex. 2. îs "Every plane section of a
paraboloidof revolutionis projected into a circle on the tangentplane at the vertex."
Ex. 3. Anyplanethroughthe Anyplanethmughthe tocmiacentrelapet'pendiMtM'to the con. perDendieulattotheNnejoiniNgthejugatediameter. focmto itapole.
Et. 4. ThéconewhoMbaseia AnytaNgentoone has for itaanyMotionof aepheteha<its oir- foeat!iM<the lineejoiningthew~-oulareectionaparallelto theplane tezof theooneto thetwofod.ofaection.
Ex. 6. Anyplane i< petpendi. Thelinejoininganypointto theodMto the linejoiningcentreto &ena1aperpendicularto thep!aneitapole. joiningthe foenate the inteKee-
<tonwiththe directrixplaneofthepolarplaneofthepoint.
MCtMOCAÏ. SCNPACM.126
Rjt.&E~tyeytMetONebp- EtetyMcttMpMtiagthMnghiag a ttpheMil <~ht. théfooMbastM<foe)Mfora foca*.
Et. ?. Anytwoeot~j~atttight AnytwocM~u~teHaeoareeaohliassaretmttaa!lypetpMMHco!M.th&ttheplanesjoiningthemto the
&WMareat righttn~tet.
Ex. 8. Aï)yqM<hicenwl<~n{{<ta If<tquadt!eenvetope(mmt&eeofepheMiea Mt&eeofTevotntton. tewintion,the <Maeecwlo~ngthe
former,whoMvertexil a &x!Mofthélatteriscone of revolution.
164. The prodnct of the perpendicalars&om thé two foci
of&ant&Mo{T<~Mt<mMtmA&etNMvetMaM€m<aytangent plane~îa evidently constant. Now if we rec!proc&tethis property with regard to any point, by the method nsedln Art. 168iwe leum that the squareof the distance from the
origin of any point on the reciprocalsm'&oeis m a conatantratio to the prodmotof the distancesof the point from two &ced
phmea.It appears fmm Ex. 4, of the last article, that the two
planes are planes of circutarsectionof the asymptoticcone tothé new surface; that Mto aay)that they are planes of emsalMsectionof the new surface. The intersectionof the two planesM the reciprocal of the Une joining the two JM; that is to
say, of the axia of the mrBMeof revolution. The propertyjust proved* Mtongs, as we know (Art. 148), to every pointon the mnbtnca.rfocal comc,hencethe reMptocatof any quadricwith regard to an mabîKcar&<~Ma,sar&~<)fMvolntI(m
rMmiAe~mM~œam~b~hMg~i~&BM~~r&~aïa a Bsr&ceof revolutionround the conjugate axh). By red-
procating properties of enrËMettof revointton, we obtain pro-parties of any quadric with regard to &OMand coneapondmgdoectnx. It ia to benotedthatinetthercaae&eaxtsof the SgOMof revolutionia the rec!procal of the directrix
coïMspondmgto thé given &)Ctm.
It watmtMaway1WMCat ledto tMaproperty,andto otMtTethedMMdonhetweenthetwttdB~ef&ei.
BECIMOCAÏ. SOBFACE8. 127
Thé aais of the figure of revolntion is parallel to the tangentto the focal corne at the given fbcaa (eee Art. Î87).
The left-hand column contains properties of etn'&ces of re-
volution, the right-hand of quadrica in gener&L
Ex. 1. The tangent cone whoM The cône who8evertex ia a focusvertes la any point on thé axis le and base any section whose planea right eone who<e tMgent planes paMe~ tbrough the CMt'etpondin~make a <!0)MtMtangle with the <tiTeettix,h&Hghtootte,whoMtade
plane of contact, wbioh plane la la the Une joining the foeu to thé
pMpeMtMtr to the axh. pole ofthe plane of section, and this
right Une ie petpendienh)' to the
plane through &Masand dtteettùt..
E:c. 2. Any tangent plane la at The Une joining a 6teu< to any
right angte*with the plane throngh point <m the anf&ee il at rightthe point of contact and the tuda. angIeato&eËMjoining~he&MM
<othe point wherethe conetpondingtangent plane meeta the dh'Mttix.
Ex. The polar plane of any The line joining a &e<Mto anypoint la at right angles to thé plane potnt la at right angles to the
eontaining that point and thé axia. Une joining the focaa to the pointwhere the polar plane meeta thedh-eetth.
Etc. 4. Any two eonju~ate Unes Any two conjugate lines pieteeare MMhthat the planes joining a plane thmugh a directrix pataHetthem to the locus are at right to OMutar sections, in two pointa
angles. whieh Ntbtend a right angle at the
correepondingtbeM.
Ex. 6. If a cone ehfcamMnte Thé cone whoaebaMMo~pTaneNn&<eof revolution, one principal aectionof a quadrio and vertex anyplane is plane of vertex and axie, foeaohas for one axis the Unejoin*and another is parallel to plane of ing to the &eaa thé pôle of thécontact plane, and for another the linejoin-
ing &oaa to the point where thé
plane meeta the direettix.
E~. 6. Thé coae whoee vertex Thé eone !a a right eone whoMM a fbcM and base any plane sec- veftM la a ibena and baae the tee-tion fa a right eone. tion made by any tangent one on
a plane through thé corteapendhMdirectrix pm-aUet to those of théeireata)' aeotiont.
MSCïHtOCALSUBFACES.128
Ex. 7. Lecue of interaection of IfthMUghMypointenaquadncthree tangent planee te a parabo- be drawn three lines mntMDy at
loid, matuajiy at right angles, la a right an~lex,the plane joining their
plane. other extiMmittet paMM through a
nMd point,If the point be not on the quadrio
the plane enve~pe< a sat&ee cf
revolution.
Ex. 8. If a quadrio envelope a If two quadriet envelope eachaurfaceof revolution, thé MMof the other, the eone, whose vertex la anylatter M parallel to a principal plane locus of one and whieh envelopesof the former. the other, bas for one axis the line
joining that jbetMtothe point wherethe plane of contact meeta thé cor.
tetpon([mg<Ureetrix.
( 129)
K
CHAPTER VIII.
CONMCAL8URPA6E8.
165. WB shall in thin chapter give an acccant of thoae
properties of surfaces which are anfdogonsto those propertiesof coNicewhich are connected with their foci, And we com-mence by pointing ont a method by which we shoutd be leito thé conaUera~onof the focal conics of a quadric, inde-
pendently of the methodfoUowed(Arts. 186,&c.).Two concentric and coaxal cornes are 8Mdto be confocal
when the differenceof the squares of the axes !a thé same for0~ O*
both. Thus given an eU!pse + H = Ï<any conmis confocal
with it whoseequationis of the form
If we give the positive sign to X', the confocalconicmU bean ellipse; it w!Nabo be an ellipsewhen Is negative aa
long as it !s tess than y. When is between &'and ?' theconfocalcurve ïs a hyperbola, and when X' is greater than a*the cnrve is imaginary. If ~~y the eqnation reducing!<se!fto ~=0, the axis of a: it&e!fMthé limit whieh séparâtes con-focal e!I!pae8&om hyporbolaa. Bat thé two &ct helong tothis limit in a spécial sense. In fact through a given pointc'y can in general be drawn two conics confocalto a givenone,anco we have a qoadratMto determine viz.
130 COSFOCALSURFACES.
the second root is <deoX'~t*, and therefore the two foci are
in a spécial sensepoints correspondingto the value ~'=f. If
166. Now in like manner two quadrics are said to be
confocal if the differencesof thé squares of the axes be the
lame for both. Thus given the elUpMtd.s + Ja + T=t any
surface is con&eàtwhoseequationia of the form
If we give X*the positive aign, or if we take It negativeand less than o*the sor&ce Man ellipsoid. A sphereof infinite
radius is the limit of ail e!lipeo!daof the syatem, being what
the equation represents when X*=oo. Wben X* ia betweenc* and b' the snr&ce M a hyperboloid of one sheet. When
it ia between b*andd' it il a hyperboloidof two sheets. When
X*=<~the auT&cereduceaIt~df to the plane z='0, but if we
make in the equationX*<=< .T=0tthe points on the conic
thus found, viz. + ~j, = 1, belong in a special sense
to the limit separating elKpsoidsand hyperMotds. In tact,in general through any pointicye' can be drawn three sarfaces
confocal to a given one; for rega~diag as the nmknown
quantity, we have evidently a cuMcfor thé détermination of
it; mamely,
CONFOCAt.SUM'ACES. m
K2
and a root of <&Mequation w!!ta!Mbe X* c*,if
The pointe on the focal ellipse therefore belong in a specialsense to the value \*=!tc*. In like manner the plane y=00
separates hyperboloidaof one sheet from those of two, and to
this limit belongs in a special sensé the hyperbola in that~t t
plane + =e L Thé focalconic in the third principal
plane ie imaginary.
167. 2~ <~Me~Ma<&*«M!oA<e~can be drawn ~~M~Aa ~Mx~point CM/~eo)?to a ~toeMone are respectitely an eM<~MOt<7,a
~~pefi~M~of one < and oneof <<co. For if we snbatttntein the cnMoof the !ast article Baccessîvely
we get results saccesstvety +–+– which proves that the
equationbas alwaysthree real roots,one of which is less than e',the secondbetweeno*and and the third between &' and <and it was shown in the last article that the 6Nr&coscorres-
ponding to thèse values of are respectively an ellipsoid,a
hyperboloidof one sheet, and one of two.
t68. Another convenient way of solving the problem todescribethrough a given point qaadncs confocal to a givenone, is to take for the unknown quantity the primary axisof the songht confocal sarfaco. Then mnce we are giveno*-& and <t'c" which we shaHcall and ?*,we have thé
equation
From this equation we can at once express the co-ordinatesof the intermtion of throo confocal surfaces in tenaa of their
CONfOCAL SURFACES.132
axes. Thns if a", a' ?"" be thé roots of the above equation,the îast term of it givea lu at oncefc"& =o''a"'a" or
And by parity of reasoning,sincewe might have taken &*or c*for our unknown,we have
N.B. In the above we supposef, y* &c. to involve their
a)gns implicitly. Thns c"" belongingto a hyperboloid of one
aheetia eesentlaHynegative,as arealao& and c'
169. The preceding cubie aho enables as to express theradius vector to the point of intersectionin terme of the axes.For the second term of it gives<N
This expfeasMnmight a!sohave beenworked out directly fromthe values given for a: y", z'*in the last srtide, by a proceaawhieh may be employedin reducingothersymmetrtcolfunedonsof these co-ordmatea. For on substitutingthe preceding values
and redncmg to a commondenominator,a/*+~* + e" becomes
But thé numerator obTlons!yvanishesif we mppoBeeither
~'=c*, <~=< <~==& It M therefore dMatMe by the de-
nomm&tor. The divisionthen !s pertbnneA as follows Anytenn, for example e"<t"*<!t"when divided by <&* (or byits equal a"&") ~veaa quotient e'o""o', and a remainder
~<t"*a'c'. This remamderdividedby o"* b' gives a quotient
TheseMp'eMhMMenableneea~tytorememberthe co-ordintteaofthenmMUM.TheumhHtMarethepointa(Art.139)wherethefocalhyperbolameetsthe mr~ce. But for the focalhyperbola<t**c a"* a* y. Theco-ordinateearetherefore
CONFOCALSURFACES. 188
yW and a remamder~<t" b~hdividedin likemanner
by a"& gtvee a quotient &"& and a remainder &y"&which!adeettoyedby another termin the dividend. Proceediagstep by atep in this manner weget the result already obtained.
170. Ï~OOCOM~M<~MM~MM<<<«tc4 other et)MyM~'6<!<
t~A<<M~&<.Let icy~' be any point commonto the two Bnrfacea, and
thé lengthsof thé perpendicularfromthe centre on the tangentplaneto eachat that point, then (Art.85) the directMn-cosinesof thèsetwoperpendicularsare
And the condition that the two shonid be at right angles,M,(Art. 18)
But mnce the co-ordinates.c'y* satisfy the equattom of bothsm~Meswehave
And if we subtract one of thèse equations from the other,and rememberthat a"* a**'=' & & =c"* c' the remainderis
whiehwas to be proved.At the point therefore where thrae confocalsmtersect, each
tangent plane cuts the other two perpendicolM-ty,and the
tangentplane to any one contains the normaleto the other two.
171. If a plane te d&MMM<X~A <~ centreparallel to any<tM)y«!tplaneto a ~t«K&TC,Me<M!Mof <ec<MMtmade &y<&)!<
jp!sK<are ~ofaMe~<o<!5eMMWMJb? <~ <!co<~):~Ma&<j4wMy&<<epoint of CMtihMt.
It hm been proved tbat the pt~raUelato the normals are at
right angles to each other, and it only rem&ioeto be proved
CONMCALaUBMCES.134
that they are conjogatedtNnetentin their section. But (Art. 90)the conditionthat two Ëmefteh<KtMbe conjugatediameteMia
But thé trath of this eqnattomappeam at once on snbtractimgone &om thé other thé eqnatMmawhich hâve been pfoved în
thé last arëdet
172. ,/M !eay<&<of <t.KMo/' thecentral <ec<MMof a
~t«MMo&ya plane parallel to the <oMj$wt)<plane <t< <~e~cMt<<c*y'e*.From the eq«nation of the surface the length of a central
radius vector whoM direction-angles are a, j8, 'y is given bythe equation
CONFOCALSUM'ACES. 186
And subatituting this value in the expreMtonalready found
&r~' weget ~<t'o" In tike manner the square of theotheraxiBB ?'<
Renée, if two confocalqutuMcftintersect, and a radius ofonebe drawnparaUetto the normalto thé other at any pointof their curveof intersection,this r&dmaMof constantlength.
t78. Since the product of the axes of a central sectionbythé perpendicahofon a parallel tangent plane is equal to <t&:
(Art. 54), we get immediately expresmons for the tengthap', p", jp" We hâve
the valuee already found for a; y", e" and reducingthe re-
sultingvaluefbr~" bythe methodofArt. 169.Thé reader will observethe symmetry'whichexMtsbetween
thèse values for j)", j~ and thé values already found for
~*) If the three tangent planes had been taken asco-ordinateplanes, wouldbe the co-ordinatesof thecentre of the surface. The analogy then between thé valuesfor p'p"p'" and those for a!'y' may be stated as follows Withthe point a; as centre three confocalamay be described
havingthe three tangent planes for prmcipal planes and inter-
secdng in thé centre of the original system of snr&ees. Theaxes of the new eystem of con&calsare a', <t",a' & S")& 7c', o",o" The three tangent planesto the newsystemare thethree principalplanes of the originalsystem.
If a central section be parallel to one of these principalplanes (thé plano of ay for instance)in thé surface to whichitis a tangent, it appoars from Art. 172 that the squaresof théaxes are a*-y, a'-c*. In other words, that the section is
136 CONFOCALaUBPACjEB.
precieelyequal to the focalellipse, no matter where the point~'bemmatetL IntikemMneftheeectionpamMtotheplane of a?~is equal to thé focal hyperbola.
174. If D be the diameter of a quadric parallel to the
tangent !ine at any point of ita inteMecttonwith a confocal,and p the perpendicularon thé tangent plane at that point,thenpD !s constant for every point on that carve of intersec-tion. For the tangent line at any point of thé curve of inter-sectionof two aar&ceais the intersectionof their tangent planesat that point, whichin this case (Art. 170)is normal to the thirdconfocalthrongh the point. Hence (Art. 172) 2?*= a"-a"
176. ?b~?K<?<~e&)CMso/jpo&<~<t~t'MM~&tMe<ct~f<y<M'dtoa f~tteMo~eoM/oca!Mt~~MM.
Let the given plane be J~+~+Cs=l, and its polethen wemuet identi~ the given e~nationwith
The locus is therefore a right line perpendicularto the givenplane.
The theoremjust proved,implicitlycentainethe solutionofthe problem, to describe&ear&cecon&calto &given one to
touch a given plane." For since the pole of a tangent planeto a s)M&oais tts point of contact,it is evidenttbat but onesurfacecan be AesctiheAto touch the given plane, ita point ofcontactbeing the point where the lome line just determined
Bieetathe plane. The theorem of this article may a!so be
CONFOC~LSURFACES. 187
etated–~The locus of the poloof the tangent plane to anyquadrie, with regard to any confocal,Mthe normal to the fireteut&ce."
176. Tb~X~an &)!p~S<&Mtfor Me~&<NK<!ebetween<~ J)OMt<of contactof any tangentplane, and its pole ?<? regard to any
<XM<~caJ'MM~ce.Let a!y)!'be thé point of contact of a tangent plane to the
surface whose axes are a, b, c; ?, {' thé pole of the same
planewith regard to the surfacswhoseaxesare a', b', c'. Thea,as in the lut artide, wehave
177. Tite <!a!Mof any <oMyeN<cone <o<t $t«K&'tcare <~MÛMKa&<0the <&MCCM!~M<t&<cAtC&can &eOfMHMt
oet~sco/'<&3COKe.Considerthe tangent plane to one of these three sar&cea
wh!ohpasa through the vertex ic'y')! then thé pole of that
plane with regard to the original surfaceMes(Art. 61) on the
polar plane of a~') and (Art. 175) on the normal to the ex-terior surface. It is therefore the point where that normalmeets thé polar plane of a; that is to say, the plane ofcontact of the cone.
It follows then (Art. 60) that the three normals meetthis plane of contact in three points, anch that each is the
pôle of the Hne joining the other two with respect to thesection of the surface by that plane. But since this is aisoa section of the cone,it follows(Art. 67) that the three normala
CONPOCAÏ,8CBFACN.188
are a systemof conjugale diametersof thé cône, and smcetheyare mutaaHyat right aagles they are its axes.
178. If at any point on a quadrica Unebe drawn touehingthe Burfaceand through that line two tangent planée to anycon&cat, these two planes will make equal angles with thé
tangent plane at the given point on the arst quadric. For bythé last article that tangent plane ia a principal plane of thécone tonchimgthe confocalsurfaceand havingthe given pointfor its vertex, and thé two tangent planes will be tangentplanes of that cone. But two tangent planes to any conedrawn throngh a line in a principal plane make equal angleswith that plane.
The focal CtMMN(that is to say~ the cones whose verttceaare any pointsand whichstand onthe focalcomcs)are limitingcases of cônes envetopmg confocal6m'&cea,and it is sttU tntethat the two tangent planes to a focalcônedrawn throngh anytangent lineona surfacemakeequalangleswith thetangent planein which that tangent Une lies. If the surfacehe a cone its
focal comc reduces to two right lines, and the theorem jnatatated m this case becomes, that any tangent plane to a conemakes equal angles with the planes containing ita edge ofcontact and each of thé focal lines. This theorem, however,will be proved independently in Chap.ïx.
179. It follows, fromArt. 177,that if thé three nonna!abemade the axes of co-ordinates,the equation of the cone muettake the form ~ta~+JE~+C~~O. To verify this by actual
transformationwill give us an independentproofof the theoremof Art. 177, and a Imowledgeof the actual values of A, J?,0will be useM to us aRerwards.
The equation of the tangent conegiven,Art. 74, is
CONMCALSUt~ACNt. t8&
Now to ttMM~Kato the three nonnats ae ftxeft,we hâve tosttbt~tnte the JJtectioB-commesof theec !me)t in the Aumalteof Art. 17, and we see that we have to substitate
180. In ordermore easily to eee the result of thie snbsttta-
~omthe followingpreHminaryfbrmn!~will be useful:
CONfOCAt.8UMACJE8.140
18t. When now we make the trana~rmtttMnadirecte, inthe Mt-hanAmde of the equation of Att 179, thé coeSdantof a~ Mfound to be
Its square therefore destroys thé nmt gfoap of terma on theother sideofthe equation,and the equationof thecônebecomes
whiohis the reqnîred transformedequationof the tangent cône.
182. As a particular case of the precedingmay be found
the equationsof the focal cônes (Art. 178) that is to say, thécone whosevertex is any point a!~ and which stands on thefocal ellipseor focal hyperbola. These anewer to thé values
a* a' bl for the square of the pnmary axis: the equa-tions therefore are
Theae equationsmight alao have been found,by forming, as at
p. 86, thé equattoas of the focal cones,and then trana&ïmmgthem as i&the iMt tn'tides.
CONFOCAt.SURFACES 141
It may be seen without dMcutty that any normal and the
correspondingtangent plane meet any of the principal planesin a point and line whieh are pote and polar with regard tothe focal conic in that plane. This m a particnlar case of
Art. 177.
188. Having aH thé necessary formule at hand, we givealao in this place the transformation of the equation of the
quadric itaelf to the three normalsthrough any point icy~' asaxes. The equationtransformedto parallelaxes becomes
',C~ 11' 1 =' !l' 111:'But the transformations
of~+~+~andof~-+~-+~are given m Art. 181. The transformedequation is therefore
at once found to be
and the quantity under the brackets on the left-hand s!de ofthe equation is evidently the transbrmed equation of the polarplaneof the point.
This equationis somewhatmodinedif the point a:'y'e'ia onthe surface. The equationtranafbrmedto parallel axes !a
CMfFOCALSDRPACE8.Ï42
the coeSMentof ~evamMiMwhilethe termeof the &ratd~ree
ïeduceto The tMM&nneAequation!sthereforep
184. We give in this place a!so the transformationof the
equation of the MŒproctJ-Bm&cewith regard to any point to
the three normalathroughthe point. The equationM(Art. 162)
(~'+~'+M'+~)'=~-4-&y+<
Nowusingthe~naulas ofArt. 179,the quantityaa~'+~y'+M'+A*is immediatelytrMM<bm)edinto (p'a!+~"y+?'"<+&'). Again,whenaV + +c*)!*is tmns&rmed,the coefficientofa~m
186. To return to the equationof thetangent cone(Art. 181).Its form proveathat all côneshavinga commonvertex and cir-
eumscribinga seriesof confocalaurfaceaare coaxaland confbcaLFor the three normalathrough the commonvertex are axes to
everyone of the systemof cônes; and the form of the equationahowathat thé dtSeremcesof the squaresof the axes are inde-
CONFOCAL8UKMCN8. M3
pendentof a'. The equations'ofthe commonjbca~mea of the
coneaare (Art. 140)
But it was proved (Art. t72) that the central section of thé
hypcrbotoidof one aheet which pasBesthrough a/ is
and the sectionof the hyperboloidby the tangent plane iteelfissimilarto this,or is aiso
Rencethej~eo!ifM:~o/ <y<<emof conesare thegenerating?MM<of the~pef&o&)t<i'icAttAp<MMN<&)'o!<~A<&e~MM!<–atheorem
dne toJacobi (<M&, Vol. xn. p. 137).Thismay a!sobe proved thus Take any edgeof one ûf the
systemof cones,and throngh it draw a tangent plane to that
cone and ako planes containing the generating Mneecf the
hyperboloid;theselatter planesare tangent planesto the hyper-boloid,and therefore (Art. t78) make equal angles with the
tangent plane to the cone. The two generstors are thereforesnch that thé planes drawn through them and through anyedge of thé cônemako equal angles with the tangent plane tothé cone; but this !sa propertyofthe &cat Imes(Art. 178).
CoB.1. The recîprocab of a system of con&eabt,with
regard to any point, have the same ch'cola)' sectiona. Forthe iec!procabof the tangent cônes from that point have thesame circnlarsections (Art. 141), and thèse rectprocab are the
asymptodoconesof the reciprocalsur6tces.
Cott.2. If a System af confocalebe projeeted orthogonallyon any plane, the projections are confocal conics. The pro-
jectionsare the sectionsby that planeof cylindersperpendicularto it, and emvetopmgthe qaadncs. And these cylindersmaybe conmderedas a system of enveloping cônes whose vertex!s the point at infinity on the common direction of their
generators.
CONFOCAL8UMAOE8.144
186. Jf~CO<'OM/M<~<M<~K!MC<M)be<&~MM? <M<C~a ~rtcealine.Take on the line any point a;y< let the Mea of the three
Mr&ceapassing thMBgh it be a', d', <t' and the angles the
Hne makes with these axes a, 'y' Then it appean, from
Art. 181,that <tis determinedby the gM<M&~<«:
have the given line as a commonedge, and it is proved, pre-
cisely as at Art. 170, that the tangent planes to the cônes
through this Uneare at right angles to each other. And sincethe tangent planes to a tangent coneto a surface,by definition
touch that surface, it followsthat <&e<ttt~<K<planes <~(KMt
at~M~Aany f~A< line to the<<COeMt/boa&tO~MA <Ot«!~M,areot right <Mt~!ettoe<K!&ot&er.
The property that the tangent cones from any point totwo mtetaectmg con&cah eut each other at right angles, issometimesexpressed as fbilowa: teaoCMt/be<t&<eeKj~'<MManypoint appear <0<K<eMeC<e«My!0&eM<!<right angla.
187. If ~Mt~r~a given lifte tangentjp&MMbe drawn to a
oy~emof o<M/!M<!&,the co~veopoK~M~K<WNM&genemtea /~per-MMJMMSoM~.
The normals are evidentlyparallel to one plane; namely,the plane perpendtcalar to thé given line; and if we consider
any oneof the confocals,then, by Art. 174, the normal to anyplane through the line containsthe pole of that plane with
regard to the assnmed confocal,which pole is a point on the
polar line of the given line with regard to that confocal. Honoe,everynormal meets the polar line of the given Unewith regardto any confocal. The surface generated by the normab isthereforea hyperbolieparaboloid(Art. 111). It is evident thatthe surface generated by the polar lines, jnst referred to, isthe same paraboloid, of whieh they form the other system of
generators.
CONFOCALSURFACES. 145
L
The points in which this paraboloid meeta the given lineare the two pomtt where this line touches con6)cats.
A spécial case occurs when thé given Uneis itsdf &normalto a surface <S'of the system. The normal correspondingto
any plane drawn through that lime ie found by letting &U a
perpendicularon that plane from the pole of thé eame planewith regard to j8 (Art. 176), but it is evident that both pôleand perpendicolarmuet lie in the tangent plane to )8'to whtchthe given tine is normal. Henee in this case aU the nortnalalie !n thé sameplane.
From thé principlethat the anharmonicratio of four planespassingthrougha Uneis the sameas that of their fourpoleswith
regard to any quadric,it is foundat once that any fournormaledividehomographicallyall thé polar lines correspondingto the
givenl!ne with respectto the systemof snr&ceB. In the spedalcase, now under consideration,the normals will therefore en-
velopea conic,whichconic willbe a parahola~since thé normalin one of its positionsmay lie at infinity; namely, when thesurface is an infinite sphere (Art. 166). The point where the
given Une meets the surface to which it Mnormallies on thedirectnx of dus parabola.
188. If a, y be thé direction-angles,referred to the threenormala through the vertex, of the perpendicular to a tangentplaneofthe coneof Arts. 179,&c., since this perpendicular liesonthé reciprocalcone,a, /3, y muetsatis~ the relation
This relation enablesus at onceto determine the axis of thesurfacewhich touches any plane, for if we take any point on~heplane, we know a', d' a'" for that point, as abo the angleswhichthe three normalsthrough the point make with the plane,and therefore«' is known.
189. If the relation of tho last article were proved inde-
pendently, we ahould, by Mvemmgthe steps of the demon-
etration, obtain a proof without transformation of co-ordinates
CMtMCAI.BCN'ACB).146
of the equation of the tangent eoM (Art. 181). The following
proofis due to M. Ohmtes Thé quantity
is the sam of the squares of the projections on a perpen-dicular to the given plane of the lines a', a", 0' We have
aeon (Art. 173) that these are the axes of a snr&ce having
iB~V for Its centre and passing through the original centre.And it waa pïoved in the Mme article that three other con-
jugate diameters of thé eame snBEMeare the radius vector
from the centre to a~ together with two unes equal and
parallel to the axes of the focalellipse. It was a!ao proved
(Art. 74) that the sum of the squares of the prûjec~onaon
any âne of three conjugatea!ameterscf a quadrio is eqaal to
that of any other three conjugated!ameters. It &!tow9thenthat the quantity
is eqaal to the sam of the squaresof the pr~ectioM on thé
perpendicularfromthé centre on the given plane, ef the radins
vector, and cf two Imes equal and pamM to the axes of the
focalellipse. The l~twoBneaaM constant m magnitudeand direction,and their projectionsare thereforeconstant,while
the projection of the radius vector is the perpendicatar ttadfwhich M constant if ~y~belongto the given plane. It!a
provedthenthat the quantity
NCMU!tantwhHethepomt~~moTMm<tg!venp!ame;an<!it is evident that thé constant value M thé <~of the sar&cewhich touches the gtvemplan~amce&rttwe have
190. ï*i5e&CMof the «!<eMec<MH<ib~e~&MMMM<M<t~at
~W~&~M~C~MM~tW~MMM~t~MMM~Mb&a~&ThMie provedae in Art. 8$.
Addtogether
CONFOCALSURFACES. 147
L2
where p is thé Satanée from the centre of the intersectionofthe planes.
Agaïn~by subtracting one from the other, thé two equations
~*=a'c<M*a+&'coa'~+c*cos*'y, F"'=a["cos'<+&"cos'~+c"cot~'y,we learn that the differenceof the squafesof the perpend!cn!<tmon twoparallel tangent planea to two con&calBia constant and
equala'-a".
191. J~coeoKMA<tMM~n ommon wy<apenvelopetwocoM-
J~M&/ <0jM &Ky<&0/' the M~C~pt NMM&on one of theircoMMMtedgesbya plane ~o<~& theceM~ejMMtNe!? thetangentplane to one of the cot~M& tSw~ Me<M!'<ea;.The interceptamade on the four common edges are of course ail equal sincethe edges are equally inc1inedto thé plane of sectionwhichia
pataUelto a commonprincipal planeof both cones.Let therebe any two oon&calcones
Pottmg in the vaîaes of et*,~8*,'y*from thé eqn&t!onaof thé
tangent cènes (Art. 186), and ïemembenng that the <c'of thé
plme throtigh the oentre isa'c**
(Art 178) weplaNethroogh thé centMis
(Art-~3), we
get for the eqaare of the required mtercept
If then thé mu'&cesbe ail of differentkundethis value shcwsthat thé interceptMequal to thé perpendicular from the contreon the tangent plane at their intersection.
In the particular case where the two cones consideredarethe cônesstandingon the focal ellipse, and on thé focal hyper-
CONFOCAL8UMACE8.148
botawehaTea*=a'-o% a' a*-y, and thé intercept reducesto a'. Henee, if throughany point on an eMtpM~be drauna chord meeting&o<A~cco~conice,theintercepton <&McA<wo'
a ~&Methroughthe centre parallel to the tangentj~<MMat the
point will 6e equal to ~e <M!M-m<M'o/' the M~~Me. Thia
theorem, due to PMf. MacCuUagh,Mttnatogonato the theoremfor plane curves, that a tiDethrough the centre parallel to a
tangent to an ellipse enta off on the focal i'adS portions eqaalto the axia-ma~or.
192. M. Chastes bas used the principlesjust estahlished tosolve thé problem to determinethe magnitude and direction ofthe axes of a central quadric being given a system of three
conjugate diameters.Conaider 6rst the plane of any two of the conjugate dia-
meters, and we can by plane geometrydeterminein magnitadeand direction the axes of the section by that plane. The
tangent plane at J~ the extremlty of the remammg diameter,will be paraUel to the tiame plane. Now it WM proved(Art. 178)that the centre of the given quadrio is the pointof intersectionof three confbcak,having the point P for theircentre. If now we couldconstmotthe focal contesof this new
system of confbcab, then the two focal cônes, whose commonvertex Mthe centre of the originalquadric, determineby theirmutual intersectionfour right lines. The six planescontainingthese four right lines intemecttwo by two in the directions ofthe required axes, while (Art. 191) the three tangent planesthrough the point P eut off on thèse four lines parts equal in
length to the axes.
The focalconics requiredare immediatelyconstmeted. Weknow thé planes in which they lie and the direction of theiraxes. The lengths of their axes are to be o'-a" <ï"-<t"<t* a", ?" a" But nowthe lengthsof the axesof thé givensection are o'-o", a'-a"* (Art. 172), and thèse latter axes
being known,the axesof the focalconicsare immedîatetyfound.
193. If through any point P on a quadric a chord be
drawn, as in Art. 191, touching two con&cab, we can nnd
CONFOCALSUM'ACES. 149
aaexpfeMion&rthelengthofthtttchoTd. DrawapM&Ueleemi*di<HMteterthronghthe centre, thé length of whichwe 6!M!1
coUJB. And if through P there be drawn a plane conjugateto thm diameter, and a tangent plane, they will intercept
(counting from the centre) portions on the diameter whoee
pfodnct='jB*. But the portion mtercepted by the conjugate
plane !< hatf the chord required, and the portion intercepted
by the t<mgentplanemthe intercept found(Art. 191). Hence
may represent a right cone, two of the coefficientsmuet be
equal; that ia to say, a"==a', or o"==< or in other words,for the point iey~' the equation of Art. 166 muet hâve two
equal roote, but from what was proved M to the limitewithinwhich the roots Ke, it is evident that we cannot have equalroots except when X is equal to one of the principalaxes, orwhen a~'< is on one of thé focal con!c<. This agrces with
what wu proved (Art. J44).It appears, hence, that the reciprocal of a surface, with
regard to a point on a focal conic, is a mtr&ceof revolution;and that the reciprocal, vith regard to an mnh!l!c,ia a para-boloid of revolution. For an amMHcis a point on a focalconio(Art. 189), and atnce it is on the surface the reciprocalwith regard to it H a paraboloid.
Another particular case of this theorem is that two rightcyundefs can be tarcanMcnbedto a centml quadric, the edgesof the cylinders being parallel to the asymptotesof the focal
hyperbola. For a cone whosevertex is at infinity is a cylinder.As a part!cu!arcase of the theorem of this article, thecône
standing on the focal ellipse will be a right cône only when
CONFOCAI.SURFACES,150
ita vertex is on the focal hyperbolaand <tMewef«!. This
theorem of course may be stated withoutany ïe&remceto thé
qattdtice of which thé two eomceare focal cornes; that <~
&)CMÛ/'<~ Wef<MM0/' ~A< CMMa«~M&<<ttK<~<Mt? ~CCM<??«!M a eoKtcq~<~<M~6apec<Mw o j)MyeM<MKt&ty~&Me. K the
equation of one conic be +==t, that of thé other will
It was proved (p. 126) that if a quadric circumscnbe a
mn&ce of revolution, the cone envelopingthe former whoeevertex M a &cm of the latter M of revolution. From thisarticle then we see that the focalconicsof a quadric are thelocus of the foci of all possibleettr&cesof revolation which
can dromMcnbe that quadric.
196. The &]lowingexampleswillserve fnrther to lUnstratethe principles whichhave beenlaiddown
Et. 1. Tofindthé loeu oftheintMMet!onof generatorsto a hyper.boloidwhieheutaat rightangles.
Theeeedonparallelto the tangentplanewhicheontaÎMthegeneMtortmn<tbean eqnilateralhyperbola,tethat(Art.l?2)(o*- o*)+(«~-a~)c0.But(Art.169)theeq~reof theMdt~veeteftothepointSa
We hâve, therefore, thé tocut a ephete, the square of whose radtiMis eqnaltoa~+y+e". OtherwiMthM:Iftwo)pmer&t<m~attH~htMtj;!jeaj.thetr plane together with the plane of eaob and of the normal ttt thé
point, Me a system of three tangent phnee to the surface, mutually at
right MgtM, whoM tateMeettonlieson thé <phete <<°*y -K* (Att. 89).
Ex. 2. To Bad the loous of the intetMetion of three tangent lines toa quadric matutUy at right angles (see p. 86).
Let a, ~9,'Ybe the angles made by one of these tangents with thenormale through the locus point, and sinM each of these tangent* lies
on the tangent cone through that point, we have the conditions
161CONPOCAh8CBFACE8.
AndtheMmofthe tMiptoeahof therootswiNVMiehwheathé eotBo~tef\'=& TM<,theMbM,~wa<M<!heeqM(ti<Mtcft!Mt<M<MMqaiMd.
Bx. S. Thé M<t!onof an eBlpMMby the tangent planeto the Mymp-tetieconeof een&etdhyperboloidb ofcomftMttatea.
The MM (Art.92) la invenelypMpOT~<m<~te thé petpendieatm en&pmaMettmjj~ntphNe,mdwehave
BatdneethepMpendi<M!Mhaned~<ecnerM;pKMltOtheMymptetieeoMo~thehypetboMdtWehtw
Et. 4. To Cadthé lengfh et the petpendieaîat&omthe eentMonthé
pohr phae ef fty~ in tenna of the MM of thé o<m&)c<ttawhichpMathKMghthat point.
196. Two pointe, one on each of two confocal dIIpsoMa,Mestndtocon'eeponAif
It is «Vtdantthat the mtersectton of two confocalhype~-MoHe perces a syatem of etHpsotdsin conrespondiagpomta,
a'a'~a'°for from~Mvalue (Art. 168)
a~t=. j[.the qnantity
N <MBBtantM long as the hyporboloUb,having «", a"*ibr
axM, are constant.
CONFOCAÏ.SURFACES.t62
It will be obeervedthat, the principal planes being limitaof confocal surfaces,pointe on the principal planes determined
a/' F «" yby equations of the form
-.=-jt–?t ~='M '~) ~*==0,
correspond to any point a/yV on a surface, and whena)'y'z' iain the principal plane, the eorrespondingpoint is on the focalconio.
1&7. The points on thé plane of «y, which correspondtothe intersectionof an ellipsoidwitha aeriesof confocalsurfaces,form a series of confocal conics, of which the points corM-
aponding to the umbilicsare the commonfoci.
EUmin&ting betweenthe equations
This is evidently an ellipse for the intemect!ons with hyper.boloidsof one sheet,and a hyperbolafor the intersectionswith
hyperboMJs oftwo.The coordinatesof the umbilicsare
which are there&rethe fbct of the systemof confocalcomca.Cnrves on the ellipsoid are sometimesexpressed by what
are caUed etitptic co-ordmatea; that is to aay, by an equationof the &nN~(a',a")'=0, expressinga relation between théaxes of the con&calhyperboloidswMch can be drawnthroughthe point. Now sinceit appenrefrom this article that a' is ha!fthe sum and «" hatf the difference of the dtatances of the
pointa correspondingto the points of the locus from the pointa
CONFOCAt.8UBFACE8. 163
which correspondto thé umbilics,we can from the equation(a', a") 0 obtainan eqaatbn (p+ p', p') =0, fromwhieh
we can form thé equation of the earve on the principal planewhichcorrespondsto thé givenlocaa.
198. If the intersection of a sphère and an ellipsoidbe pro-jected on either plane of <atcntarsection by lines para1leltothe least (or greateat)axis, the projection will be a orde.
This theorem in only a par<ico!ar case of the ~bHowmg:that "if any two quadrics have common <arcnlaraectionf),any
quadrio through their mtetBectMnwill have the same;" atheorem which M évident, sinceif by making e'aOm !7 andin F the resnit in each case représenta a cirele, making ee=00in !7''tA~ mast a!so represent a cirele.
It will be neeM, however, to investigate this partîcnlartheorem dh'eetty. If we take as axes the axis of y which is
a line in the plane of ciredM*secdon and a perpendicular toit in that plane, the y will remain unaltered, and the newa~=i the old a~+~ But by the equation of the plane of
.j, c* «'-y. the .j. &' ~c*.0?.otTcataraection
:==-?“a-,the new
ar=-.ar.
o o –F or y–<rBut for the intersectionof
CONMCAt. SMM.CM.154
x*–«*h(~etathMeofthe!atterthee«BBtactMtto–s–.
Henoe
wenMyhBmedMttotyin&r&oïntheïtMtmtïcteth&ttheptû-jection of the intersection of two con&caî q<M<Mcs<ma,plane of oiroalar section of one of them Ma conic whose fociare the NBuhH'pK<ject!otNof the maMHea;and, &ga4 that
given any carve (a', a") on the ellipsoidwe eaa obtain thé
a!geb)"MCequationof the pro)eot!<mof that enrve on the phaeef <mcntfu'eection.
199. 2~ <S<<<!MMe<~e<«)eH~<«?pointe, OMM each of tw
<!<M< eNt~MOM&& egW<~to Me<N<&aM<~e<<M8Mtk <!Mcorre-
epMM~~MK~.Wehave
Thé mm of the squaresthereforeof the central radii to thétwo points Mthe sameas that for thetwo conesponding pointe.But the quantities a~ yl~ &~are evidentlyrespectively equal
to a~'Z', y, e'F', sincejr~~?,«!' &c. The theorem<t
of this artîde, due to S!r J. Ivory, ia of use m the theory efattractions.
200. h order to obtain a property of quadries <matog<t<Mto the property of con!cs&at the snm of the focal distancesMconstant, Jaoobi states thé latter property as follows Takethé two points 0 and C' on thé eU!pseat the extremity of the
axis-major, then the same relation ~+p''=2<t which connectathe dietancesfrom <7and 0' of any point on the line joiningthese pointe, connectsaiso the distances&omthe foci of anypoint on the ellipse. Now, in Ncemanner, if we take on the
CO~MCAt. SURFACES. 1M
principal sectionof an ellipsoid thé thMe pointe which corre-
epond in the eeneeexplained (Art. 196) to any three pointaon the focaleBipM,the eame relation whichconnectathé dis-
tancée from the former points of any point in their planewilltJoo connectthe dMtanceefrom thé latter pointe of any pointon thé Mn&ce. In &et, by Art. 198, the dititanoeftof the
points on the contboalconic from a point on the eNr&eewill
be eqnal to the distances of the point on thé principalplanewhich c<wvMp<M<&to the point on the Mu'&ce,B'omthe three
pointa in the principal section.*
201. Conversely, let it be required to find the !ocnaof
a point whosedistances from three nxed points are connected
by the same relation sa that which connectathe distancéefrom
the verticea of a tnangle, whoseaides are a, e, to any pointin ita plane. Let p, p', p" be the three distances,then (Art. 50)the relationwhichconnectathem M
Bat p'–p") &c. being only fanct!onB of the co-ordmate* of
the 6rst degree, the locM is mam&stly only of the eecond
degree.That any of the points from which the distances are
measnred M a focus M proved by ahewing that this equation
Mr Townsendhallthewed &om geomettiMl eoMidet~tiom (CbmtW~w«tMtJOxt~ Jt.Mt<mo~M<Jo~<fMo~Vot. m., p. JM) that thia property onty
Mo!~ to pointaon the m«!t<~<'focal cornet, and in fact the pointa in thé
planey whieh correspond to any point <V<' on an ellipsoid are imaginM'yM etaly appeMa from the formula of Art. t9C. Mr. To~Mend easilydet4re<Jacobi'a mode of ~jeneratton &om MMCaHagh'a modular pmperty.For if through any point on the aui&ee we draw a plane pMaHetto a
etteahr section, it wiU out the diteotir!e<aconeoponding to the three
&Mdfooi in a triangle of invMtabte magnitude and Ngate, and the distancée
of the point on the mnAoe &om the three foci will be in a constantratio to its dieiMeee &om thé Tettices of thit triangle. And a Mm!t<o'
triangle oan be formed with ito aides ioeteaMd or dhninMMd in a mxed
ratio, the dhtfmcM &om the Tert!cea of which to the point <'y~ sha!l be
equat to its dbttuoea from the foci.
CMtMCALMJKPACES.M6
Mof the fbrm ~S+M<, where 8 mthe Mnitety amaHepheMwhose centre is this point. In other words, tt ia rM[a!redto
pmve that the result of makingp'~ 0 !n the preceding equationMthe product of two equationsef the &'st degree. But thatresult is
where ia the angle opposite« in the triangle a~c. But this
br~supmtotwouiMgmMy&ctom,BhMnngthat<;bepo!ntwe are dmcnsstngîe a &ooaof the modular kind.
M~MMM~~M~M~p~M~O~~MC~
<MK~CCt&,<~ &M<Mof ~e&'jpOM~of COM&tC<M <t ~p~M)kLet a, 7 be the doect~on-tmgtesof the perpendicular on
the tangent p!ames. Then the direction-cosinesof thé radius< a'cosa &'cos~ o'coe'yvector to any point of contactare i
~p'<y)y«
ae easily appears by substituting in the formula (Art. 8S)
oof)<t'=' <*cos<t'for a!'and solTingfor cos<t'. Fonamg then
by Art. 15, the du'ect!on-cos!neBof the perpendtcnlar to the
planeof theradius vectorand the perpendicalaron the tangentplane,we findthem tobe
where ia the angle between the radius vector and the pep-pendical<u'.Now the denominatorM double the area of the
triangle of which the radius vectorand perpendicularare aides.
Double the projections,therefore, of this tnangle on the co-ordinateplanes are
(&<) c<M~cosy, (<<~) cos'y Ms<t, (<t'-y) coaetcos~.
Now these projectionsbemg constant for a tystem of confocal
saï&cea,we learn that for each a eyeteat, both the plane of
CONFOCAÎ.MIM'ACES. M7
the triangle and tte magnitude is constant. If then CM be
the perpendicularon the oeriesof parallel tangent planes andPM thé perpendicularon that Nnefrom any point of contact
P, we have proved that the plane and the magnitude of the
triangle CPM are constant, and therefore the loeu of P ie a
hyperbolaof which C~f is an Mymptote.
208. The Mc!pfocalof a systemof confocalsnt&ces
Now the latter equation denotesa system of quadricspassingthmugh a commoncurve, one qnadrio of the System beingthé point aphere a~+y'+~'c'O. The reciprocal system is
therefore inscribed in a commondevelopable. Many of the
propertiesproved in this chapter for confocalMM&ceacan bedenved as particular CMeaof properties of snr&ceainsenhedin a common developable. Compare Arts. 182, 170, andArts. 122, 176.*
Since thé tangent cone from any point on a focal conicieone of révolution that is to say,one which haa doublecontactwith the imaginary cirele at infinity(Art. 1S5),it followsthat
throughany point on a focal contecan be drawn two imaginaryplanes which will touch every confocalsurface, and we thus
aeegeometricallythe existenceof this developable,the tangentplanes to which toach ail the eon6)ca!s. And we can ataosee that it ia the same aa the developable generated by the
tangent planes to the sartaoe whiehpasa through the tangentato the imaginary cirde at infinity. The actual equation of
the developableia obtained by fonning thé discriminantwith
regard to of the equation of the oomocab. The imaginarycircle at infinity and the focal conicsare all double limesonthis sor&ce.
Seeabo C~ade~J9M. Secrn.,p. 397,and QtMW~J~~~6!<~otM<&<,Vol.m., p. t«&
158 COttVATOM0FQU~MtïCS.
CUBV~TUBE<? QUADRIC8.
204. The general theory of the cttrvatnre of mrfaces willbe explained in Chap. x., but it will be convenient to statehere some theorems on thé curvature of quadncs which are
immediatelyconnectedwith the aubjectof this chapter.a normal <ec(t<Mtbe madeat <M~point on a quadric its
~tradius of CMnM<Mye<!<<&<!<point is e~MNt~to M~C/3Mthe
MBtwMM)te<efparallel to the &'<Meof the «c~MKon the tangent~p~MM,<MMfjp M<~ pMy<Nj<CM&M'J~'<Mttthe centreon <&<<OK~!<plane.
We repeat the followingproof by the method of infini-teshnata from CMtMa,p. 296, which see.
Let P, Q he any two points on a quadric; let a planethroagh Qparallel to the tangent plane at P meet thé centralradius CP in JB,and thé normal at P m then the radiusof a circle through thé pointa P, Q having ita centre on .P~
JPÛ*gis But if the point approachmdenmtètynear to
QP is in the limit equal to ~JB; and if we denote CP andthe central radins parallel to C~ by c' and ~3,and if F* bethe other extremity of thé diameter<X~then (Art. 70)
a" jRB.~P' (= 2a'R)3j3*J'B j8' J°R
therefore QJB'== and the radinsof carvatUM='r. nB.erelore =<t
an t e lUS0 curvature<t ~.o
But if 6'om thé centre we let fati a perpendicular CM on tha
tangent plane, the right-angled triangle OMP ia similar toJHM and ~B PB a' And the radius of carvatnre is
pt pttherefore a. S- whiohwaa to be proved.
p pIf the cirele through PQ have its centre not on PB but on
any Une P<S"making an angle with JRS,the only changeFO*
is that the radins of the cirde îa being ~till on the
plane drawn throngh Q paratld to the tangent plane at P.
CCBVATUNBOf QCADNCS. 169
BntJRS'evidenHy~JRS'oM~. Theradimaf cm-vatmeMjpû*
therefore coa~ or <~ value for de MxJ~Mo/' eMtx~MM
~M~!M~~&<~twMM<~a~M~Mc~~e~MM~<ee<<Mt<&w~AJP~, mM~a~ by oos9.
SOS.These theoremsmay abo easily be proved analytically.It Mproved(CbM~ p. 806)that if ~!B*+8& <y+2~=0 0be the equation of any conic,thé radins of curvature at the
yorigin M="~
If then the equation of any quadric, the plane
ofa!ybeinga tangent plane,be
then the rattU of curvature by the Mcttom ~=0. <e'=0 areJ! y
Kspecttvety-r,Batif the equation be transformed to
paraHelaxes through the centre, the terme of highest degreeremain analtered, and the equation becomes
jy tr
ThesqaaresoftheiBteKeptsonthefHdBofœand~a.M-This proYMthat the radii of curvature are proportionalto the
squaresof the parallel aemt-diametersof a central section. And
ince, by the theory of conics, the radins of curvature of thatsectionwhich contauN the perpendieularon thé tangent plane
Mthe same is the form of thé radiuaof every other section.
The same may be proved by aNBg the equation of the
quadriotransformed to any normal and thé normale to twocon&cahae axes (Art. 188),vu!.
ThemdSofc<!rvatnMofthe secttonaby the planes <f==0,y=00
n mM'eMSpecdve!y–~L,–I~L.Thennmeratorsaw the
Fs
psquaresof the eemi-axes of the section by a plane parallel tothé tangent plane (Art. 172). The equation of the section
CCBYATURBOPQPADB!C().t60
made by a plane making an angle 6 with the plane of y is
found by 6rst tuming the Mtee of co-ordmatearound throughan angle C by sabsttttttingy coaC–~ e!n~, y sm~+fi!coaC for
y and < and then making the new ~=0. The codBdent of
y*will then become an /1
But this coemcientof ia evidently the square of that semi-
diameter of the central section,which makes an angle 0 with
the axisy.
206. It followsfrom the theorem enunciated in Art. 204,that at any point on a eeH(f<!<!guadrio the f<K?MMof CMfMtMfS
of a tMntM~section &<M<tmaximum and minimumvalue, the<RMe<«MM< the <ec<«M!for ~eM values M~ paraliel <o<&e
<t!BM~M~-and aa!M-M:'MM'of the central MotMnby a plane
parallel <0the tangentplane.These maximum and minimum values are called the ~MtM-
<~p< radii of cnrvatnre for that point, and thé sections to
whiclr theybelong are called thé principal sections. It appearafrom (Art. 171) that the principal secdons contain each the
normal to one of the confocalsthrough the point. The inter-section of a qnadrio with a combcat is a curve snch that at
every point of it the tangent to thé cnrve is one of the prin-'
cipal directions of curvature. Such a cnrve is ca!led a line
of ourvatureon thé surface.In the case of the hyperboloidof one sheet the central
section is a hyperbola, and the sections whose traces on thé
tangent plane are parallel to the asymptotesof that hyperbolawill have their radn of carvatnre infinite; that is to say, theywill be right Unes,as we know aheady. In passing throughone of thosesectionsthe radius of carvatnre changes sign thatïs to say, the direction of thé convexity of sections on oneside of one of those lines is opposite to ~at of those on theother.
CUBVATUMEOP QCAMtCS. 161
M
207. 7!eojMW)c~MJoeM~et~e«HXt<<<~<tM~t!M~M Me<<M~<M<~&NM«<? regard <0<&!<<M<!Mt)M'«?)&<«wMe~pass tih~M~~the ~CMf contact. For these pôles lieon the normal to th&tplane (Art. 176), and at distancesfrom
it = and (Art. 176), but thèse have been just
provea to be the tengths of the pnndpal radii of eorvatnie.We can ako hence 6nd, by Art. 176, the co-ordinatesof
the centres of the two principal cu'dea of corv&tnte,v!z.
808. If at each point of a quadric we take the two pna-cipal centresof cnrv&tnre,the locus of all thèse centres Masorfaceof two theets which is càUed the Moj~eeof ceM<~M.To find its equation, we observethat the co-ordmates<c', e'
mtts~ the equations
Sobstttatmg for <c*in terma of a: by the help of the hst
article, Mtdwriting for a", a*-A*, &e., we obtain the follow
ing two eqn&ëons:
Theeoequations express that all the centres which correspond~pM~<m&&M~o~~mMon~e~MmM~M~&r~M&&MCM~M~ËeomABU~mM&mofhMqw~M& If weelimmateA*between thèse two equations, we get the equationofthemmfMeofc'iHitMS. IhM~pM&mMdt~~mm~&ma~g~mtheK~M~M7~m~~VoLU! p. 318.~ The smf&ceis one of the tweK)hdegree.
&maybeworthwhiletoetttetheprecemby whichtheeliminationWMe~Mttd!1
CU&VATUBE OF QCAONCS.t62
209. We can Me&pn<Mtthé nature of the sectionof theoar&ceby the principal planes. Im faot, one of the principalradii of curvature at any point on a principal section M the
tadma of curvature of the section itself, and the locus of the
centrea correspondingis evidentlythé evolute of that section.
The other )t~)M of curvatureoorreapondingto any point in
the aectionby the plane of a:y!s as appeam from the for-
mtda of Art. 204, sinee c is an axis in every section drawn
throngh the axis of e. From the &rmnhe of Art. 307 the
co-ordmatesof thé correspondmgcentre are a: -T– y';
that is to aay, they are thé potes with regard to the focalconic of the tangent at the point a:y to thé principal section.The locus of the centres will be the reciprocalof the principalsection,taken with regard to thefocalconic,viz.
The section then by a principalplane of the sur&ce (which isaf the twet&hdegree) condstsof thé evoluteof a eotuc,whichia of the s!xth degree, and of a conic (it will be found)three Hmeeover, this coniobemga double Ime on the surface.The sectionby the plane at infinityis a!soof a mmihrnatare.
wMe the right-hand aides of the other two équation: are got by writingy aad in tara, inatead of a*in this lad equation. We thm get threelinear relations between P, <~ j~ & But fhrther, fînee these quantitieaare MeN<i!entaof a biquadratio equation which bas three roots equal,those eoeSciente are eonnected by two Mtat!otN, one cf the second, the
other of the third degree. The elimination ia thaa teduoed to e)!minatMn
betweena cubie and a quadratic equ<tt!cn,whioh Mpmet~cabiB.
163CORVATUBE0F QUAMïCS.
M2
210. The t~WMaï of <j~«M~tceof eeK<Me& a «o~ace<<~e~CM~iO~)'<M.
It will appear &omthe general theory of the curvatureof
snr&cea,to be explainedin the next chapter, that thé tangentplane to either of the con&c&laar~cea through <c'y'<'is ~bo a
tangent plane to the surfaceof centres. The rec!proca!sof the
interceptawhichthe tangent plane makeson the axes are givenby the equation
But it !s evident (asat .B%~ .P&HMCt<n!M,p. 14)that
may be understoodto be eo-ordinatesof the rectprocalsurface;amce, if be the co-ordinatesof the pole of the tangentplane with regard to the Bpheïa a~-t~+<=!, the equation<~+~+<{'= 1 being iden~calwith that of the tangent plane,
Cwill be atM tho reciprocalsof the mtetceptBmade bythe tangent plane on the axis.
( 164 )
CHAPTER IX.
CONBSANDMHBM-CONiœ.
211. IF &coneof any degreebe eut by any sphere, whosecentre is the vertex of the cone, the cnrve of section will
evidently be suchthat the angle betweentwo edgesof theconeM DMMttMdby the arc joining the two eotteapontUngpo!ntaon thé sphere. When the cone ia of the aecond degree, theeorve of eeo6onMcalled a <)pAew-cotMe.By atating many ofthe propertiea of conesof the seconddegree as propertiesof
sphero-conics,the analogy between them and correapondingpcoperdes of comcebecomesmore stnkmg*
Stnctty speaking, the intersectionof a aphere with a coneof the <t'°degree !a a carre of the aa**degree: but when thecône Mooncontncwith the ephere, the curve of intersection
may be divided, in an infinity of way~ into two aymmetncaland eqnal portions,either ef wHchmay be regarded as analo-
gons to a plane carve of the M*~degree. For if we conaiderthe points of thé cnrve of intersectionwhieh lie in any hemi-
sphere, the points diametricaByopposite evidently trace out
a perjteetlysymmetricatcarve in the oppositehémisphère.Thua then a spherc-cMnemay he regarded as anatogooa
either to an eBipseor to a hyperbola. A cone of the second
degree evidently Intenecta a concentricsphère in two sanihu*`
closed ourves diametricattyopposite to each ,other. One ofthe principal planesof the cônemeets neither eurve, and if welook at either of the hémisphèresinto which this planedivides
See M.ChMte~eMemohonSpheto-wntM(paNMMdin the SxthVolumeofthe 7hMM<M<&<M~f<it<J~ ~«!<tt)Myof J~yMM&,and<MM-!*tedby PtotiMtMGtavM,Dublin,183T),&ûmwbkhtheNMmdattMMef
manyofthe theetenMmthieohapteraretaken.
CONB9ANBSPHEM~CONtCS. 165
the sphere,we seea dosed ourveanalogousto an ellipse. Butif we look at one of thé hemiaphereainto whiohthe sphere
ia divided by a prinoipal plaue meeting both the oppositecorvée, weMe&carveconsistmg of two oppoaite brancheslike a hyperbola.
The cnrve of interaecdonof any qaadnc witha concentric
t~heM ia evidenûya apharo-eonic.
212. The propertiesof epherical corvée havebeen studied
t~DMMN~q~mM~q~MMde~M~M~e&mMdondMm<~<~ Cartes~ o<W(K!mates.ChooM~MMofcc'-Md!-
natMMtytwogre&t oircles OJS', OFintemectmgatnghtangles, an~ on them let M perpend!calM'ejHt~ PN &'omanypoint on the sphere P. TheMperpen(~calN'9aMnot,asimplane oo-ordmates,equal to the opposite sides of the quad-rilateral OJMRy; and therefore it would aeemthat there in
acertaîn!&titmdea~!m!b!emoarMlecdoncfephencalco-otdînatM, according M we ohooae for co-ordmatea the per-pm~M~Ml~wt~m~~b<M~6~vM~th~make on the axes.
M. Chtdermamiof Clevesbas chosen for oo-ordmatesthe
tangents of thé mtercepts 02!~ ON (see CreUe'eJ~tHM~VoLVI.,p. 240),and thé readerwill findan elaboratediacM~onof dus systemof co-ordinatesin the appeodm to Dr. Gh'atres's
tMna!at!<mofChadea'aMemo!ronSpheM-<iomcs. Itiseaayto see however that if we draw a tangent plane to the sphereat the point 0, and if the lines joining thé centreto the pointsJM,.S, meet that plane in points m,o, then ÛM,0<twi!lbe the Ca!'tesianoo-ordinateaof the point p. But <?M,OMare the tangents of the arcs OJM,ON. Hence the equationof a sphencat cttrve m Gndennann'e system of eo-ordinates!s in reality no&ing but thé ordinary equation of the planeearve in which the cone joining thé sphencal carve to thécentMof the sphere is met bythe tangent plane atthe
point 0.80 again, if weohoosefor co-ordinateathe einesof the per.
pendioobmPM,PN, it is easy to see in like mannerthat the
equationof a spherica1corve in saeh co-ordinatesis only the
CONESAND6PHEB<~CONM&166
equation of the orthogonalpro)ec6onof that curve on a planeparatlet to the tangent plane at thé point 0.
It seems, however, to as tbat thé properties of ephencatenrves are obtained more simply and directly from the eqaa-tions of the cones whichjoin them to the centre, than &omthe equations of any of the plane curvea into which they cam
be projected.
2] 8. Let the co-oNtmatesof any point P on the sphere besnbstituted in the equation of any p!ane passing thmugh thecentre (whichwe take for origin of co-ordinatee),and meetingthé aphere in a gréât circle ~J9, the resatt will be the lengthof thé perpendicalar from P on that plane; which is the sineof thé tpheneal arc têt Ml perpendicnlarfrom P on the greatciro!e AB. By the hetp of this principle thé equations of
cones are interpreted so as to y!etd properties of sphencalcnrvea in a manner pMeieelycorrespondingto that med in
interpreting the eq~tatioMof plane curves.
Thns, let a, j8 be the équations of any two planes thronghthe centre, whieh may a!aobe regarded sa the equationsof the
gréât cirolesin which they meet thé sphere, then (as at CMt«w,
p. 5S) a– A~denotes a gréât oMe sach that the sine of the
perpendicular arc from any point of it on « is in a oonetant
ratio to the sine of thé perpendicularon that is to say,a great cirde dividing thé angle between a and /9 into partswhose aines are in the same ratio.
Thus, again, a– a-i!j8 dénote ares fonning with a&
and ~3a pencHwhose anharmonic ratio iap. And <t-~
a+~3 denote arcs formingwith e, ~9a harmonie penciLIt may be noted here that if be thé middto point of
an arc AB, then B, the fourth harmonieto .A', and J!, ia
a point distant from A' by 90°. For if we join these pointsto the centre <7,CA' is the internai bisector of the angle ~C~and therefore CB' muet be thé externat bisector. Conversety,if two correspondingpoints of a harmonie system are distantfrom each other by 90°, each is equidistant from the other two
points of the system.
CONES AttD SFHERO-CONtCS. 167
It !e coovoniontako to mentionhere that if <c'y'<'be thec<Mrdmateaof any point on the sphere, then aa/+tM'denotes the great oirde having a! for its pole. It is in&et the equationof thé plane perpendicularto the line joiningthe centreto thé point a!'y'<s'.
214. We can now !mmed!<tte!yapply to ephencat trianglesthe methods<Medfor plane triangles (C~«a, p. 54, &c.). Thusif a, /3, Y denote the three mdea,then, M in plane triangles,?<t'!=)M~c:t!Ydenote three lines meeting in a point, one ofwh!oh pMeesthrough each of the vert!ce9 while
are thé mies of the triangle formed by connecting the pointewhere each of these joining lines meets the oppositesides ofthe given triangle; aad ~t+M~+M-ypasses through the inter-sections of correspondingsidesof this new triangle amdof the
given triangle.The equations e:.=./3=<yevidently represent thé three M-
sectors of the angles of the triangle. g~nd if .4, 0 be the
angles of the triangle, it is easily proved that as in planetriangles a cos~ = cos~t<=y cos<7 denote the three per-pendiculars. It remains tme, as at CbK«M,p. 5A, that !f the
perpendiculaKfrom the vertices of one tnangle on the sidesof anothermeet in a point, so will the perpendicularsfromthévertices ofthe secondon the sidesof the first.
The three bisectors of sides are a smJ.'=j8 amJS='y sinC.The arc (tsin~+~8sinJB-ysm<7 passes through the three
points where each a!de is met by the arc joining the middle
points of thé other two; or, again, it passea through the
point on each side 90° distant from its middle point, for
Ctsin~i~sinB meet y in two points which are harmonie
oonjagateswith the points inwhioh a, meet them, and sinceone is the middte point the other mnst be 90°diatant from it
(Art. 218).It followsfrom what bas been jast said that the point
where asin~+~8smJ9+'ysiaC meets any side is the poleof the great circle perpendicular to that side, and passingthrough its middle point, and hence that the intersection of
CONESAND 8PHBM~<!Ot!tC6.:?
t&e<h~mohperpand!<adMe;thatistoMty, the centrerthe ciroumecribingoircle is thepe!eofthe gM<tte!rc!eae!N~+~a!aF+'y~n<X
216. The conttittonthat two gréât drctes <!?+ ~y+ <w,
Ot'ip+t'y+c'~ ehouldbe pcrpetMticatfH'is mani&stty
<M'+M'+«!0.
The condition thttt <Mr-t-6~+.<~~a+~+c'<y aho<tMbe per-peadMa!aria easily foundtrom this by sttbtttMntimgfor a, ytheir expMMtonain term of a~ e. The resolt !s exaetly theBMMaefor the correspondingcasein the plane,vis.
Imlike mannerthe eineof the MeperpendicniMto <Mt+~+<)and passingthrough a given point is found by sabaHtatmgtheco-OKtiaateeof that point in oM+~+ey and dividing by the
square root of
8M. Passing now to eqaationsof the second degree, we
may consider the equationay='<M~*either as denotinga cone
having e and <yfor tangent planes, while /3 passes throughthe edgea of contact, or as denoting a epheto-comc,havinge and <yfor tangents, and ~3for their arc of contact. The
equation phunîy aosertathat the produet of the aines af pef~pend!cnlamûorn any point of a sphero-conioon two of its
tangents is m a constant ratio to the square of the sine of the
perpendicularfrom the samepoint on the are of contact.In uke manner the equationa'y~~M assertB (see CiMtMe,
p. 81~) that the product of the sines of the perpendicularsfrom any point of a aphero-comMon two aides of an inscribed
quadniaten)! is in a constantratio to the product of aines of
perpentScttlamon the other two mdea. And from this pro-
perty again may be deduced,precîaely as at C~MMw,p. S16,that the anharmonicratio of the four aMSjoining four fixed
points on a sphem-eonicto any other point on the carve is
constant. ln I!~e manner ahnost all the proo& of theoreme
CONESAND aPHËBO-CONIOB. 169
respeeting plane conica(given Ctm~, chaps.XïV.,xv.) apply
eqnaUytoepheK~cm~ee.
217. If «, Il represeat the planes of drctttar aeotioa (orc~c!tc~!tHMe)ofaoone,&eeqM~enofthecomMoftheform a)'+~'+<6*'='ta~ (Art.$9), which interpreted, as in the!aat artM!e,ehewa that the productof the aineaof perpen-d!cnlam&omany point of&6phero-con!c on the tw')<yd!eam is constant. Or, again,that, GIventhe baMof a aphe~cdtt~oagi~and the product of cognéeof mdes,the locnaof vertexis a BphM'o-comc~the <yc!icaroa of which are the great <aM!ea
having for their poleethé exh'emîtîesof the given base." Theform of thé equation shewsthat the cyd!c areaof aphero-conicaBManalogoueto the asymptotesof plane coaica.
Every property of a spherc-cornccan be <toaHedby con-
mdonDgthe Bphero-comcformed by the cone reciptwat tothe given one. Thus (Art. 141)it wu proved that the cyclioplanea of one cone are perpendicular to the focal Imes of thé
reclprocal cone. If then the pointa in whieh the focal finesmeet the sphere be called the foci of the sphero-conic,the
property estaM!ahedm this article proves that the productof thé sines of the perpendicnlamlet fan from the two focion any tangent to a sphero-conicis constant.
218. If any great c!rctemeet a apheM-conicin two pointsQ, and the cyclic arcain points then~Fe=J3~.This ia deduced &om the property of the last artide in
the same way aa the oorrespondingproperty of the planehyperboIaNpfoved. The ratio of the aines of the perpen-dicalars from P and Q on a ia eqnal to the ratio of the e!nea
ofperpendtoa!M'a&om<j'andjPon/ Bat theNnesof
theperpendlcal&ra&om P and~ on a are in the ratio
e!n~P:s!n~ amdtherefore we bave
NnJF: einAQ am-B~ mnjR~
whenoeit may easily be in&Medthat ~Pe.JS~.
BeciprocàUy,the two tangents from any point to a sphero.coniomake eqaal angles w!th the ama joining that point tothetwo~oL `
OONESAND 8PHB!tO-CONÏCa.170
219. As & part!cn!ar CM$of the theorem of Art. 218
~Ieanith<tt<~FM<MM~*<<ay &t<tj)~<<'a<pZt<w~MMOM~W~&~MM&W<CM~ &&M~M~M~
o~<iOK<(M<.Thia theoremmay alM be obtainedby the me&odof m&oi~Nma!9from that of Art. 217 or it may be obtamed
<t!recttyfrom the equationofa tangent, T!&
The &rm of this equationshewsthat the tangent at any pointis constmoted by joining that point to the intersection of ita
polar (axc'+~'+M', sec Art. 218) with a'~+~ot* which isthe
fourth harmonie to the cycliearcs a, /9, and the line joiningthe given point to their mtersection. Since then the givenpoint is 90' distant from its harmonie conjugate in respect ofthe two points where the tangent at that point meets the
tydic arcs, it Mequidistantfrom these points (Art.218).
Beciprocatly, the lines joining any point on a sphero-conioto the two foci make equal angles with the tangent at that
point.
220. From the fact that the intercept by the cyclie arcson any tangent îs bisectedat the point of contact, !t may atonce be inferred by tho methodof in&titesima!s(see CbMM~,
p. 294)that et'ay tangenttoa apAet~-cMtM~M'aMM!~ &!ecye?M<tMwa 6'«tM~ of CMM&M~area,or a triangle the snmof whose
base angles is constant. This may a!so be in~Med trigono-metricaUyfrom the fact that the prodnct of sines of perpen-dicnlare on the cyclie arcs is constant. For if we ca!t the
intercept of the tangent c, and the angles it makes with the
cyclic arcs and B, the sines of the perpendicalars on aand ~3are respectivelysin~cs!n~[, sin~<s!njR. Bat consider-
ing the triangle of whicho is thé base and and B the base
angles, then by spherioattrigonometry,
But 0 !s given, thereforejS,the half snm of thé angles,is ghren.
B~M~~<~<Mm~Mj~M~o/~t~~y~M~Me~MMCMeM~M~ Or the same may bededuced by the mBthodof m&utesunats (see C~M«w,p. 297)
CONESAND SPHEtKMJONMS. 171
tfum the theorem that the foesl radii make equal angles withthe tangent at any point.*
221. Convereely,ag<nn,we can find thé Io<MMof a pointon a sph~M,Mtok that the sum of its distaneeafrom twonxett points on the sphere may be constant. The equationooB(p~p')c:ooB<tMay be written
If then « and ~3denote the planes whieh are the polara ofthe two given points, eince we have a~cos~, the equationof the locneM
In orderto prove that the planes a and/9 are perpendicularto focal lines of thMcone, it is oniy necessMyto shew thatsectionsparallel to etther plane have a &etNon the line pe~pendiootarto it. Thus let a', a" be two planes perpendicnlarto each other and to a, and therefore pasang through theline which we want to prove a focal line. Then Nnce
If then this locus be eut by any plane parallel to a, a'ta"*ts the square of the distance of a point on thé section from
the intersection of «'a", and we see that this dMtance is in a
constant ratio to the distance from the line in whieh jS-a oosa
Here again we can Me that a spbero-eonio may be regarded eitherMan ellipse or hyperbola. The focal lines eMh evidently meet the spherein two diMnetticaBy oppoaite pointa. If we ehooM for foci two point*wilhin one of the clo<ed ourvet in whieh the cone meett the aphere, then
the «tm of the focal distances i< constant. But if we Mbttitate forone of thé focal distances FP, the focal dMtMMejh)m the diamttnefttlyoppodte point, then since JF"P IW jRP, we ahould have the <f~MtMeof the focal distances constant
In like manner we may oay that a ~MtaHe tangent mekes wîth the
eyetieatM angles whose differenoe is constant, if we mtNtitnte ita supple-ment for one of the angles at the beginmng of thie article.
172 CONESANC 8PHENO-CONM9.
McntbythcMmephae. TMtNneiBtherefoMthediMetnxof thésection,the point a'a"beiag the &etM.
We eee thm abo that the général equationof a conehaving~e!imea~&ra&caHimeisofthe&)rma~+~'=(o!f+&y+0!)';wheBceagain it followathat thasineof <~ dSM&MMeof <Myp<w<<<? <f~M-«Mt& ~M <!J~C!M&W a <!MM<<tM<ratio to ~e <MM
~<~ oM!<<M)ceo~<~e«MM~oMt~OBta oertaind'M~!<!marc.
222. ~[t~ <Mo<MtM<~&<<Mye~MMe<<~ o~eSboo~ Mtfour~ow& to~M&lie on a c~<!&.For if Z, ~f be two tangentsand 2! the chcfd of contact, the equation of the sphero-coniomay be written in the form JMf=jB*; but this mast be iden-tical with <~=a!'+~*+e'. Henee <~8-.Z~f is identical with
<e'+y'+)!jB*. The latter quantity représentaa smaUcMe,having the Mmepoloae B, and the form of thé other shewathatthat cirde oiMmMcnheathe qnadrilaterale~Zif
RedprocaHy,the &eat rad!i to any two points on a aphero-conic form a sphencalqnadnhteMdin which a sm&Horde cambe inacribed. From this property again may be deduced thétheorem that the snm or dKSH'eneeof the &cal radii is con-
Btamt,since the differenoeor aomof two oppositesidesof aucha quadrilateral ia equal to the diffemee or enm 01 the re-
tnaitottg two.
228. From thé properties just proved for conea ean be
dedaced propertiesof quadnoa in general. Thns <~ ~Mw&M<of tha aMMsof the <tN~&t«5<!t«My~<KeM<M'of a ~pe~~oM~makes <0t~ thaplme of CtM«&M'MC<«MtMeMM<<!H<.For the
generator is parallel to an edge of the asymptotiocône whose
<arca!aj sectionsare the Mmeaa thoae of the surface. Again,since the focal lines of the asymptoticcôneare the asymptotesof the focal hyperbola,it followafrom Art. 820 that the mm
or dMerence N constant of the angles which any generator ofa hyperboloidmakes with the asymptotesto the focal hyper-bola. A~M~~MMMMœea~M~~t~ayM~tha NMMor J~<~ & ~tcett of <&<angla <e7K<!&<!?~&tKCMa&Mw&&<J!ejp&MM<o/' <!?<(&?'<ec<MM!.For (Art. 98) givenone axis of a central sectionite plane touchesa côneconcydio
CONES AtfD smNMMXMMCS. Î79
with the given quadric, and therefore the pKsent theoremibBovBttefnee&'mArt.MO.
We get an MpMas!onfor the Mmor ~MSaf~nceof theangles,in terma of the given MOB,by coBmdMmgthe principal Mo-tîon c<mt~iBgtheg)'e<tteatand!ea6t)Mtes of the quadric.Weobtain the cy~cphneeby Meeting in that section
som-diametem OB,OB' each a t.«
Théo the planes containing theeeï!mea <m<tperpen<KcaIa.rtothe
plane of the figure are the cyc1icplanes. Now if we draw anyse~UMetef~'BMMt~MtMtgbawithO~wehaTe
Buta* MohvMnsty tmaxMoftheMctMnwhtchpaœeathrough it and ia perpendicular to the phne of the tigare,Md (if a' be greater thon b) a !a evidently ha!f the aum ofthe angles BOd', J~O~' whichthe plane of the sectionmakeswith the cyc1ioplanée. If a' be less than OA' Ma between
OB, OJ9',and a iBhatf the dMa-enceof BOA', JS'0~ Butthia aamor <HSeMnce!s the eame for all sectionshaving thesame mdB. Hence, if a', Il be the axes of any central aectîon)making angles C'with the cycUeplanes,webave
M~~M~MM~W~M~H~Mmb~<~<MM~<Z<!ea<MJ'MC<MKMpWp<M~MM<~<? t~j~M&<C< <i~eMM~~d~eM&WM~&F~M~
224. We mw (Art. 818) that given two ephoM~omo!having the same <yc!ic arcs, the mtetcept made by thé outer
174 CMfE9ANDSPHERO~OXtCS.
on any tangent to theinner ÎBbisectedat the point of contactand hence, by the method of Innnitesimate,that tangent eutao~ fromthe outer a segmentof constant area (CMtMt,p. 294).
Again, if two ephero-contcahave the Mme &)ci, and if
tangenta be drawn to the inner from any point on the outer,these tangents are equally inelinedto the tangent to the outerat that point. Hence, by h)&ntea!ma!s,(aee C<«w, p. S97)the excesa of the anm of the two tangents over the includedarc of thé inner conicia constant. Thia theorem ia the l'ecî-
procal of the 6rst theorem of thm article, and it is so thatït was obtained by Dr. Graves (aeehia translation of Chastes'a
Memoir,p. 77).
22& ?b Me locusof M<e!'MC<MHof <WO<<tK~K&?o <pi5eM-eoKMM~M~outat ~A< angla. ThMia in other wordato find the cone generated by the mteraectMnof two rect-
!B' W* S*angniar tangent planeato a given cône + + = 0'
Let
the direction-anglesof the perpendicatarsto the two tangent
planes be ct'~y, <!t"~8"'y";then they Mm thé relations
cos'a'+~eos'+Ccos'y~O, cos'a"+J?c<M'+CcoaV=.0.
But if a, 'y be the direcdon-cosinesof the line perpendicularto both, we have o<)t~at=l–<!0f"<t'–cos* &o. There&)re
adding the two precedingequations, we have for the equation
ofthelocoa,
&cone concycUcwith the reciprocal of the given cone. Recb-
procaUy, the envelopeof a chord 90*in length t8 a aphero-
conic, con&càtwith the reciproctd of the given cone.
226. To~M? the &C!M~'<& of ~~ye!!<MMf&!rj~OM<X<j~Mo~ <t~XeM-eMtM<Mt<Xetangent. The work of thie
question is precisely the fMuneas that of thé correspondingprobleminplane comcs,and thé only dMEsTenceis in the inter.
pretation «f the resalt. Let the equation of the aphero-conic(Art. 831)be a:'+~==<' where <=<M!+!y-)-<?,then the equa-tion ofthe tangent is t 1 jt
CONESAND SPHEBO-CONïCf). t75
is not, Met p!<NM,an identîcal relation satisfied by thé perpen-dicakrs from any point. It remains then to Mk how thethreeperpendicularsfromany point onthree fixed great circlesare connected. But this question we have implicitlyanswered
already, for the three perpen~McIaKare each the complementof one of the three distancesfrom the three polesof the sidesof the triangle of référence. If then <!t,6, c be the sideB;jd, C the angles of the triangle of réfrénée, then /9, ythe aines of thé perpendiculars on the aides from any pointare connected by the followingrelation, whieh Monlya trans-formationof that of Art. 62,
-t-2~yainJ9mnCco9<!+8'y6{mnC6!n~co8&+a<t~Nn~B!njScoao
=l-CoM–C06'.B-COtf<7-3c08~COS.Cc08<7.
The equation in this form reprettentsa relation.between thesuMBof the arcs represented by a, If we want to geta MktMmbetween the perpendîcatamfrom any point of the
aphereon the planes represented by a, ~8,y, we have evidentlyonlyto multiply the nght-hand side of the precedingequationby y', and that equation in «, 'y will be the transformationof the equation a~+y'+<=i~.
Hence, it appoam that if we equate the left-hand aide ofthe preceding equation to zéro, the equation will be the aameM a~+y'+~=0, and there&te denotes the imaginaty cirdewhich is the intersectionof two concentric epheres; that is to
say, the imaginary circle at infinity (see Art. 136).
CONBStAND 8PHNM)-OONM8.176
228. This equation enabteous to nnd the equation of the
aphere inscribed m&ghnsn tetfahedrom, whooeiweeafu'e
«,/?,-y,& If through the centre tbree planes be drawn
parallel to ft, j8,y, the perpendicularson them 6om any pointwill be <<<< TheeqaattonofthesphetCMtheretbre
829. Thé equationof a ama!lcirde (or right cone) !aeaaîy
expressed. The sine of the distanceof any point of thé cMe
&om the polar of the centre ia constant. Hence, if etbe that
polar the equationof thecMe is e~==coa'~(<c'+y*+~.AHsma!!cMes then being ~ven by eqn&tiotMof the form
~=~, their propertiesare al! cases of thèse of CMucshavingdoublecontactwith the same conie.
The theory of invariants may be applied to smaHc!tdes.
Let two circtes <8"be
177COXE6 &St) WHERO-CMttCS.
X
where D is the distancebctweenthe centres.Now the correspondingvainesfor two cMes in a planeare
Hence, if any invariant relation betwecn two ctrcleBin a planeis expressed as a funetion of the radit and of the distance
between their centres, the corresponding relation for circtes
on a sphere !s obtained by substituting for r, )' 2); tanr, tanf',and secr secf' 8m2).
Thus the condition that two ch'des in a plane ahould touch
is obtamed by forming the discriminant of the cuMc cqaation,andt8e!therjP=Oor~="'±/. The corresponding equationtherefore for two circles on a sphère ts
Agam, if two circles in a plane be the one inscribed m,the other c!rcumBcrIbedabout the same tnangle, thé invariant
relation is fulfilled 0"=4AO', which gives for the distance
between their centres the expresaton Z)'=.B'-2j~
The distance therefore between the centres of the inscribed
and c!rcnmscnbed circles of a spherical triangle M given bythe formula
So, in like manner, we can get the relation between two
circles inscribediB)and oreuNMcnbedabout thé aamespherical
polygon.
280. The equation of any BmaUcircle (or right corne)intrilinear co-ordinatesmuât (Art.227) be of the &)rm
CONESAHfDSPHEBO~ONtCS.178
!MtTe&[+<K/3-t.)t'y-<tam.~+~s!n.B-t<yfimC'.Hence,Mwas
proved be&re, this represents the polar of the centre of the
circumscribingcirele. Substituting thMvalue, the equatiooof the 8m&!lcircle becomes
The equation of the inscribed cirele toms out to be of
exactiy the same form as in thé caae of plane triangles, v!z.
( 179 )
N32
CHAPTER X.«
GENERAL THEORT O? SUBFACES.
INTBODUCTOMCHAPTER.
281. BESERVtKG for & ûtture c6apter a more dettuted ex.aminatton of thé properties of Mr&tces m général, we ahaBin this chapter give an account of such parts of thé gênera!theory as can be obtamed with least trouble.
Let thé général equation of a snr&ce be wntten !n thé &nn,
where «, meana the aggregate of terma of the seconddegree,&c. Then it is evidentthat «. consistsof onoterm,«, ofthree,
«, of six, &c. The total number of terma in the equationM
therefore thé sam of ?+1terma of the series 1, 8,6,10, CM.,
The number of conditionsneeessaïyto determinea mr&ee
of the M*"degree is one less than this, or =-
The equation above written can be thrown into the form
of a polar equation by writing coeef, p cos~, p coa'y, for
whenwe obviouslyobtain an equation of the degree,which will determinen values of the radius voctor answeringto any assigned values of the direction-anglesa, 'y.
OEMBALTBEOR?0FSCRFiCËS.180
232. If now the origin be on the surface,we ha.ve <t,'=0,and one of the Mute of the equation is always ~=0. Bat asecond root of the equation will hep=0!f<t,ybe con-nected by the relation
Nov multiplyingthis equationby p it becomesjEi<!+<~+D<!=0,and we aee that it expressesmerely that the radius vectormustlie in the plane«, =0. Noother conditionis neeeasaryin orderthat the radius shoutd meet the eur&ce in two coincident
points. Thus we eee that in general through an o~MNMof
point on <!<M<tCeme CtM<~MtMan M/&t!'<yof radii MC&VM<P&M&MtK<&<Mmeetthe <M~~KMin twoCOM<CM&K<~<W!& <Ao<
M <0«!y, an t~t*~ of tangent~)M<to the M<~C6/ and ~~MKKM ? all in MMplane, called de <()tt~e~~&MM,t~MMttMt?
by <~ <'gtM<tOM?~=0.
238. The aec<<oaof any <M~<!emade by a &Mgfe)t<planeMa <!MM)e&tCMtythepoint of contactfor a <&?<&&~X)t'M~
Every radiusvector to the surface,whiehUes in the tangent
pbme, is of comse a!so a radius vector to the section made
by that plane; and aince every snch radins vector (Art. 332)meets thé sectionat the origin in two coincidentpoints, the
origin is, by definition, a double point (see JS%r~- J'%MM
<~CM, p. 27).We have aheady had an illustration of this in thé case
of hyperboMds of one aheet, which are met by any tangentplane in a coniobaving a double point, that is to say, in
two right lines. And the point of contact of the tangentplane to a qnadrioof any other species is eqnaUyto be considered as the intersectionof two imaginary right lines.
From this article it follows convenety, that any planemeeting a snr&oe in a cafve having a double point touchesthe sat&oe, the double point being the point of contact. Ifthe section have two doublepoints, the plane will be a double
tangent plane; and if it have three double points, the plane
TMtKmai-k,1believe,WMNMtmadeby Mf. Cayley:Gr~M-y'a&?! a~M)M&y,p. M3.
OENEBAt.THEOMT0FBCtUPACES. 181
will be a triple tangent plane. Since the equation of a planecontains three constants, it is possible to detennine a planewhich will satisfy any three conditions,Mid thereforea &tHenumber of planes can in general be determined which wulmeet a given surface in a curve havmg three double points:that is to My, <tMM~tce~<M ~enow!'a <&<et~tM!o<sMtMt~a'
~<t~& tangentplanes. It wIMabo hâvean mBmtyof aouNe
tangent planes, the points of contact lying on a certain cnrvetocaa on the sat&oe. The degree of this corve, and thenumber of triple tangent planes will be sab}ect3of investi-
gation hereafter.
284. ~nb~M~Aan <!MM)M«~point <M0 MM~~ A & ~MM~~MMt&&<0<?MtM<tM?MM<MÂ~Aajh!~ <&e~meet the Mtt~ce
<&~eOOMCt&K<jpCMt<N.In order that the radius vector may meet the surface in
tbree coincident points, we muat not oniy, as in Art. 382,have the condition fulfilled
For if these conditionsworofnMHed, being a!readysupposedto vamsh, the equation of the n'" degree whtch detennmeabecomesdivisible by p% and haa therefore three roata e=$.
The first condition expresses that the radius vector must liein the tangent planeM,. The secondexpressesthat the radHM
vectormnst lie in the surface«,<=0, or
This snrface M a cone of the second degree (Art. <?) anAemce every snch cone is met by plane passmg throngh ita
vertex m two right l!nea, two right lines can be found toMnl the reqnïrea condîtion~
Every plane (bemdethe tangent plane) drawn througheither of these lines, meets the snrSMOin a section havingthe point of contact for a point of inflexion. For a point of
inflexionis a point, the tangent at wMch meets the curve
Qt!t)B!!AL THEOBY 0F SDBFACES.182
ia three coincidentpoints (~Aef ~%MM<7t<rpM,p. 85). OnthMaccoontwe shall eau the two lines which meet the surfacein three coincident pointa the Mt/&!CKMM?tangenta at thé
point.Thé existenceof these two linesmay be otherwiseperoe!ved
thns. We have proved that the point of contact is a double
point m the MctKmmade by the tangent plane. And it bas
been proved (R~~ jP&tMCxt-Me,p. 98) that at a double
point can aiwaye be drawn two lines meeting the eection
(and therefore the surface)in three coincidentpointa.
285. A douMepoint may be one of three d!S!M'entkinda
accordingas the tangents at tt are real, coincident,or !mag!naïy.Accordinglythe contact of a plane with a mr&ce may be ofthree kinds accordingM the tangent plane meets it in a section
having a node, a cuap, or a conjugate point; or in otherworda according as the inâexiontdtangenta are real, coincident,or unagmary.
Dupin, whonmt nodced* the dinerencobetweenthesethreekinds of contact, stated the matter as fbUows Suppose thatwe confineour attention to points so near the origin that all
powers of the co-ordmatesabove the secondmay be neglected,then the tangent plane (or a very near plane parallel to it)meets any eur&ce M,+M,+M,+&c. in the same section inwhich tt meeta the quadrio M,+«~ And according as theaectionsof this quadricby planes parallel to the tangent planeare ellipses, hyperbolas,or parabolas, 80 the section made bythe tangent plane is to be considered as an infinitely BmaU
ellipse, hyperbola, or parabola. This infinitely antaU section
Dapm calls the Md<c<t<Wa;at the point of contact,and he divides
the pointe of the surface, according to the nature of the in-
dicatrix into elliptic, hyperhoHc,and parabolic points. Weshall presently show that there will be in general on everysurface a nmnber of parabolic points &)mHnga corve locus,thia carve separating the etiiptic from the hyperbolic points.
SeeDipm'sjMoe&tppeoxMtftde<?&)M~<f~,p.48.
aBNBNAÏ, TttEOMr 0F BCBFACES. 189
If the tangent phme be zaade the plane of et and the oqa~'tMmoftheo!ht&oeb6
ttia marnât that the origin will be an etUp~c,hyperMIc,of pMftb«I!cpoint acoordmgas JB*M l6a!, greater th<tn,or
eq~to~O*
286. Knowmg the equation of the tangent plane when
~Mc~~M<m<~m~M~~Mc~M~mM~m~MMK~&ee~n~dMtM~mt~MMatMypoint. It Mproved ptedaety ae &t (Art. 68) that this eqa<)f.Hon may be written in either of thé forma
~')~+~')~,iu iy-r+
$I-SI (17=0,
or<?o" d~. <?jy
~+,~+~
287. Let it be required now to find the tangent plane ata point, tnde6mte!y near the origin, on tha am&ce
We have to aappoBe<<y*so small that their equareamay be
neglected; while, Nnoethe cansecattvepoint ia on tha tangentplane, wehâve ~=0: or, more M<mrately,the equation ûfthe sar&ceshows that < is a quantity of the same order asthe squaresof a!' and y'. Then, either by the formulaof thelast article, or elae diïectiy by putting iB-t-a; y+y' for teand y, Muttaking the linear part of the tnms&trmedequation,the equationof a consecutivetangent planeNfoundto be
TMeh MmetnMo expKMedas &Ho<M When the phne nf ~y h the
tangent plane, and the eqnathm of the Mt<Me i* expMSted in the tean
('t y)*w< !MW M cHipttc, hyperbotic, or pMabaBopoint Me~diag
M("')
!eM, pe<t<Mthm, or eqat! ?~) ~–).
It wiU be&nnd
that thia h eqa!vat<t)t to the ettttement in tha text: but we do net enter
into detaih bee<HMewe thttUhMa MMemoccasion in pMetîce to deal with
eqnttiMMwhere < hgiven eïpHeMy M a fanetton of arand y.
Q&NNBA~TBEOBY0F SURFACER.184
Now (<eoC~M«!Art. 141)(~a:' +~') a:+ (Rc' + <~) denotesthé diameter of the conic~+2jS~+C~*=~ which te con-
jugate to that to the point a~ Henoe any <OM~eK<plane M
<K<e!we<et?by a eonaecutivetangentplane in tho diameterof theMM&Ct<na!<C&M~& conjugateto (~ <~M'eC<MM<0tP~MAMe con-<eoM<t~e~OMtM <aJ!!eK;
This in fact ia goometricallyevident &om Dnpin'a pointof view. For if we admit that the points consecutiveto the
given one lieon an inSnitelyemaUconic, we see that thé Km"
gent plane at any ofthem willpassthrough the tangent une to
that conio; and this tangent line ultimately coincidea withthe diameter conjugateto that drawn to the point of contact:for the tangent tine ia parallel to this conjugate diameterand
inSnIteIyaloseto it,Thua then aH the tangent lines which can be drawn at
a point on a surfacemay be dmtribntedinto paira auchthat the
ta.ngentplane at a consecutivepoint on either will pasa throughthe other. Two tangent Hneaao related are called conjugate
<a~ea<i?.In the caae where the two inflexionaltangents are reaî,
thé relationbetween two conjugate tangents may be otherwisestated. Take the mnexion~tangents for the axes of a?and y,which ia equivalent to making A and <7==0in the preceding
equation: then the equationof a consecutivetangent plane ia
<+2.B(a''y+y'<c)==0. And since the lines y, o'y+y'a?,t'y -y'<c form a harmoniepencil, wc learn that a pair of
CMt/M~<t<e<Q;)!~eK<9J~fM,with thetM~aCtOM!~<eM~N«~a Aat~MKM
~CMC~.
238. In the case where the origin is a parabotic point,thé equation of the surface can be thrown into the form
'e+~+&c.=0, and the equation of a consecutivetangent
plane wiU be <!+ 2~y'y =0. Hence the tangent plane at <!Mfypoint consecutiveto a paraboliopoint passes through the in-
Sexional tangent; and if thé consecutivepoint be taken inthis direction soas to havey'==0, then the consecutivetangentplane coincideswith the given one. Hence <&etangentplane<!<a ~C!MfMtC~M~<M ? lie caMK&~ as a double ~~<K<
QMjBKAt. THEORT 0F SttS'ACM. 186
plane, sinee it touchesthé surface in two coMecutiTepoints.*In thie way parabotic points on surfaces may be cotnajaredM analogous to pomte of inSexMmon plane carves; for wehave proved (jBt~A~Plane C~fMt, p. 8C)that the tangentlimeat a point of inflexion is in like manner to be regardedM a double tangent A &!rthw analogy betwéen parabolio
points and points.of inflexionwill be afterwM~astated.It M convenient to have a nama to d!st)ngtUBhdonble
tangent planes wMch touch in two distmet points,&omthosenow under considerationwhere the two points of contactcoin-c!do. We shall therefore call the latter <&t<MKa<ytangentplanes, the word expreMmg that the tangent plane being
mpposed to move round as we pass from one point of thésurface to another, in this case it remains for an instant in
the same position. For the same reason we have ca!iedthe
tangent Unesat points of inflexionin plane curves, st&tionatytangents.
239. If on trans6Htnmg the equation to any point on a
surfaceas originwe have not only«.'='00 butaho ait the termein «, ~0, so that the equationtakes the form
.~+j~+~'+2J5~+S~+2j~+M,+&C.~
then it is easy to aee in like manner that everyUne thronghthe origin meets the corve in two coincident pomts and the
origin is then caUed a <&)!<&&point. It is eMy to aee abothat &line through the origin there meeta the surfacein threecoincidentpoints, provided that its direct!oB-cos!neasatm~ thé
equationr cos*a +F coa'~ -)-û coa*'y
+2Zrcoa~3ooa'y+2~rcoa'ycoac[+3Zcoa<[<!0!~S'=0.
In other words, <~o!<yAa double~o«!<on o! <M~MecandhttMtan t~M~ of lines MXM&<ctK<K<e<<~e<M<~eewtthreec<MKC&&t:<?<'<?< and these will all lie ona ccMof the <ec<MMf
<~ee whoseequationis =' 0. For&er, of thèse linesNXwiH
1 M!eM this WM &rat pointed out, Cambridge and Dublin J&<Ae-
M<!<M< ~«MM~,Vol. m. p. 4&
&BN&RALTHEORY Or SUMACStM
meet thé sar&ce in four coïncidentpoints; namely, the lines
of intetMcdonof the côneM,with the coneof the third degree
«,.0.Doublepointtton saï&ces nught be dasamed accordingto
the namber of thèse lines wtneh are real, or acccofdingas two
or more of them coincide,but we sMt not enter into these
(tet<u!s. The onlyBpeoMcasewhieh it Mimportantto mentionis when the COBB«, Teao!vesitself into two planes; and this
again iadodes the etiU more specM CMOwhen theM two
glanes <}o!nc!de;that is to say, when «~is a perfect square.
240. Every plane drawn through a doublepomt may mGneeemtebe regarded <M)a tangent ptane to thé sar&ce, since
it meets thé Baï&eein a sect!onhaving a donble pomt, but
in a epedal aMMethe tangent planes to the coneM,are to be
regarded as tangent planes to the surface, and the sectionsof the surfaceby these planes will each have the origin as a
cnsp. To a double point then on a surface (whieh !a a pointthrough which can be drawn an infinity of tangent planés),wil! in general correspondon the reciprocal surfacea planetouching thé saïtace in an infinity of points, wh!ch will in
général lie on a conic. If however the double point be of
thé spécial kind noticed at the end of the last ardcle, therewill correspondto It on thé Kolprocalsurfacea doubletangent
plane havmg two points of contact.
241. The resaltsobtainedin the precedingarticlesby takingas ontorigin the point we are discassing,we shall now extendto thé casewhere thé point basany positionwhatever. Let usSrst remind thé reader (seep. 29) that since the equations of a
right line containfour constants,a nnite number of right lines
<!an be detemuned to MB! four conditions (as, for instance,to touch a sni&ce four tunes); while an infinityof lines can
be 6)and to satisfy three conditions(as, for instance,to touch
a snj&ce throe tunea),those right lines generating a certain
aurface,and thoir points of contact lying on a certain locu.In a subsequent chapter we shall return to the problem to
determine in general the number of solutionswhen four con-
aBNBM. THMRY OP SCJtfACEtt. 187
ditiom are given, and to detenmM the degtee of the mt&ce
generated,aodofdtelocasof points of contact,when three
conditions are given. &t&ischapterwecoBnneoune!vestothe case when thenghtIineMreqniredtopassthroaghagivenpoint, whe&eron the anr&ce or not. This îa equivalentto two conditions; and an infinity of right line. (forminga
cone) can be drawn to satisiy one other condition; while amoiteBomber of right Unescan be drawn to aatidy two otherconditions.
We UBoJoMhmMtal'emethod employed,Ci)M< pp. 8t,184; .B%~efT~aMC<Mn~,p. 61 and at p. 47of tha volume.If thé quadriplanar oo-otdm<ttesof two pointebe a:y<'M'')a)'y<<o", then the points in which the line joinmg them iseut by thé antfaeeare found by enbstitutingin the eqnationof the saf&oe, &r Xa!'+/<.c",for y, ~'+/<y", &o. Theresult will give an equation of thé ?"' degree in /t, whoserootswill be the ratiosof the segmentsinwhichthe linejoiningthé twogiven pointsis eut by the sarfacsat any of the pointswhere it meets it. Andthe co-ordinatesof any of the pointsof meeting are ~'+~'a: xy+~t'y", \'<+~ X'M'+~'to",whereX' ~t' is oneof the rootsof the équationof the M**degree.AUthiswill présentno dMcalty to any reader whobasmastered
thé correspondingtheory for plane curves. And, as in planeearves,the result ofthe substitutionin questionmay be written
188 QBNE&AÏ.TBEOBTO?aOMACES.
the eeeonApolar, and M on: the polar plane of the same point
being
Each polar Mrtace ismani&attyalsoa polar of the point <cy<<p'with regard to all the other polamof higher deg)'«.
If a point be on a snr&ce all its polam touch the tangent
plane at that point for the polar plane with regard to the
surface is the tangent plane; and this must also be the polarplane with regard to the aeveral polar surfaces. This mayalsobe eeeaby taMngthe polar ofthe origin with regard to
where we have made the equation homogeneonsby the in-troduction of a new variable w. The polar <nu&ceaare got
by d!fE~en<Ia&!gwith regard to this new variable. Thus the
Bmtpotaris
and if ~<a0, the termeof the nret degree, both in the sat&oc<mdin the polar, w!Il be w,.
242. Ifnow the point aiy.s'w'bo onthe surface,P' vtmiahes,and one of the roots of thé equ&tiomm X will be ~tc'0.A second root of that equation will he ~==0) and the Unewill meet the Bni&tcein two coincident points at the pointai'y'tc', provided that the eoeaîctent of YNush in the
equation re6an'ed to. And in order that this should be the
case, it Mmanifestlysn~dent that a!'y~"<e"should satM~rthe
equation of theplane'y the
It ia proved then that all thû tangent limesto a sar&ce whichcan be drawn at a given point lie in a p!ane whose equationia that just written. By subtracting from thia equation, the
MtentHy
GENERAI.THEORY0F SURFACEft. 189
243. The right line will meet the surface in three con-
Mcndvepoints, or the equation we are consideringwill havefor three of Ita roots /t =0, if not only the coefficientsof X*and
\t vanMi, but aiso that of X~ that ia to my,if the linewe are considering not only Hea in the tangent plane, but
also in the polar quadric
Now (Art. 241) when a point is on a surfaceall its polamtouch the snr&ce. The tangent plane therefore,touchingthe
polar qaadno, meets it in two right lines, real or imaginary,whichare the twomBexionaïtangentsto the sur&ce. (Art.834.)
244. ~n~A<t~<MK<<M<tM<<c<M<~<~fKca(M+8)(<t-8)<[Ht~en<<M~M~tM~ot&otouch <M)~!oee&e!c<!ef&
In order that the line BhouHtonch at the point a:ys'<e',we must, as before, have the coe&denta of X*and \~=0;in conséquenceof wMch the eqttatton we are conidering bc-comeaone of the (M–8)~ degree, and if the !ine touch thesurfacea second time this rednced equation mnst have eqnalroots. The condition that this should be the case involvesthe coenMentsof that equation m the degree M-8; one term,for instance, being (A'!7'. !7)* By considenng that term weaee that this discriminant involves the co-ordinatesiey~'ie' inthe degree (n-2) (<t 8), and ayjw in the degree(tt +2)(M 8).When therefore <<'y'<w*ia Sxed, it denotes a Nu&ce which!s met hythe tangent plane in (a +2)(~ 8) nght tinee.
MNNtAt. THEMtY OP SURFACES.190
Thus then we have proved that at any point on a BM~eean infinityof tangent Unescan be drawn that thèse in generallie in a plane; that two of them pass through threo consecutive
points,and (M 2)(n 8) of them touch thé surfaceagain.
245. Let us proceednext to consider the case of tangentedrawn throngh a point not on the surface. Since we havein the preceding articles established relations which connect
the co-ordinatesof any point on a tangent with those of the
point of contact, wo can, by an interchange of accented andnnaccemtedletters, express that it is the former point whichis now supposedtobe known, and the latter songht.
Thus for example,making thia interehonge in the equationof Art. 248, we see that thé points of contact of all tangentMnea(or of all tangent planes) which can be drawn through
a;ys'M', lie on thé nrst polar, which ie ofthe degree (~ 1) ~iz.
And since the points of contact lie ako on the given surface,their locas Mthe carve of the degree ta(n 1~, which M the
intemectionof the BNt&cewith the polar.
246. The assemblageof the tangent lines which can bedrawn through a!y<'tp*forma cône, the tangent planes to whiehare aiso tangent planes to the surface. The eqnation of thiscone is &mnAby forming the discriminantof the equation of
the M"*degreem (Art. S41). For thia discriminantexpressesthat the line joining the fixed point to a~p meets the surfacein two coïncidentpointa; and there&re a~to may be a pointon any tangent linethrottgh a/y~'M'. The disonnunantis easilyae<mto be of the degree ta(M–l), and it Motherwiseevidentthat this muet be the degree of the tangent cone. For ita
degree is the same <mthe number of lines in which any planethrongh the vertex entait. But such a plane meets the surfacein a carve to which K(M 1) tangents can be drawn throughthe nxed point, and these tangents are atso the tangent lines
whichcan be drawnto the surface thyoaghthe given point.
GENERAL THEORY 0F SURFACES. 1M
247. ~n~w~Aa point not «Mthe <Mr/!M6coMin j~~M~a~<&M<cM?(?–!) (M-3) Mj~KBMK~<<M~i«. We hâve seen,Art. 248, that the co-ordinateBof any point on an mCexional
tangent are connected with those of its point of contact bythe relations 0~=0, A!7''='0, A*<7'e=0. If then we considerthé a:y«o of any point on the tangent M known; its point ofcontact M determinedas one of the intersectionsof the givensurfaceU, wMchis of the < degree, with its Smtpolar AU,whichia of the (M-1)* and with the secondpolarA' which
!s of the (N-2)~. There are therdore n (<t-!)(<2) mchintersections.
248. Through a point tMttoa the N«)~MeMM6t general i&e
<?MttOK~M(M-1) (M 2) (n- 8) doubletangenteto it. The pointeof contact of sud) Unes are proved by Art. 244, to be theintersectionsof the given smr&oe,of the nrst polar,and of the
Mtr&cerepresented by the discriminant dMCMMdin Art. 844,and whtohwe theresaw contained thé co-ordinatesof the pointof contact in the degree (M–2) (M–8). There are therefore
M()t-*l)(M-2)(ft-8) points of contact: and sincethere aretwo pointe of contact on each double tangent, there are ha!fthu number of double tangents.
Thus then we have completed the dMOtsstonof tangentUneewMch pass throngh a given point. We have ehown thattheir pointa of contact lie on thé intenection of the Bor&cewith one of the degree K-l, that their assemblageform acone of the degreeM(<t-l): thatM(M-l)(tt-2)ofthemare
inne~donat,and ~t (w 1) (M 8) (? S)of themare donMe.
TheM latter double tangents are also plainlydonbleedgexof the tangent cône, since they belong to.the conem virtne ofeither contact. Along sach an edgecan be drawntwo tangentplanesto the cone,namely, the tangent planes to the sar&ce
at the two contacts.
The mnMdona!tangents, however, are abo to be regardedas doubletangents to the Nr&ce mnoethelinepaasingthroughthree comaeeativepointsis a double tangent in virtoeof joiningthe Smt and second, and also of joining the secondand third.
The innesdonaltangents are therefore double tangents whose
SENEBAt TBfEORT 0F SURFACE.192
points of contact coincide. They are <heM&)redouble edgesof the tangent cone; but thé two tangent planes along anyauch edge coincide. They are therefore caspidal edges ofthe cone. We have proved then that the tangent coneM~M~« û/' the degree<t(M-l) bas M(K-1)(K-2) ct«p~a! e<~es,and ~N(M–l)(M-8)(t!-8) double«~ that Mto say, anyplane meets thé cone in a section having auch a number of
cuapa snd snch a number of double pointa.
249. It !aprovedpreciselyas for plane eurves (HigherPlane
CMn. page 67),that if we take oneach radiusvector a lengthwhose reciprocatis thé ?"' part of the sum of the reciprocalsof thé n radii vectoresto the surface, then the locus of the
extremlty will be thé polar plane of the point: tkat if thé
point be on the surface,the locus of thé extremity of the meanbetween the reciprocakof the n radii vectoreBwill be the
polar quadric, &c.
By mterdtumgmgaceented and ymaccentedtettem in the
equation of the polar ptame,it is plain that the locus of the
poles of all planes which pass through a given point is théSrat polar of that point. The locns of the pole of a planewhich passes through two fixed points is hence seen to be a
carve of the (M-l)* degree, namely, the intersection of thetwo first polars of these points. We see a!so that the firat
polar of every point on the line joining these two points must
pass through the same curve. And in like manner the first
polars of any threo points on a plane determine by their in-tersection (?–1)" pointa,any one of which is a pole of the
plane, and throughwhiehpoints the nrst polar of every other
point on thé plane must pass.
260. t~rom thé theory of tangent Unes drawn through a
point we can in two waysdérive the degree of the reciprocalsur&ce. First; the number of points in which an arbitraryline meets the reciprocalia equal to the number of tangentplanes which can be drawn to the given mt&ce through a
given line. Considernow any two points and B on that
line, and let C be the point of contact of any tangent plane
GENERAL THEORY 0F SURFACES. 183
passing through AB. Then since thé Une ~dC touchesthe
surface,Clies on the firatpolar of A; and for thé aamereasonit lies on the nret polar of B. The pointeof contactthereforeare the intersection of the given snr&ce,wMch M cf thé
degreewith thé two polar sar&ecs,whiehare e&chof thedegree(tt 1). The number of points of contact,and therefore<Xe
degreeof the MCtpnMo~M n (n 1)'.
261. Othorwise thos: let a tangent conehe drawnto thésurfacehaving the point A for its vertex; then emceeverytangent plane to the surface drawn through ~4 touchesthia
cone, the problem {s,to &td how many tangent planesto thecono can be drawn through any Ime AB; or if we eut theconeby any plane through jB,the problemis to find howmanytangent lines can be drawn through B to tha sectionof thécone. But the dass of a curve whosedegreeis n (K-1), whichhas <t(M-l)(M-2) cuaps, and ~!(~-l)(a-a)(K-8) double
points is
GeneraBythe sectionof the reciprocalsurfaceby any planecor-
responds to the tangent cône to the original surface throughany point. And it MeMyto seo that the degree of thétangentconeto the reciprocatsurface(aa wc!lMto the originalenr&ce)
through any point is of degree n (K-l).
2M. Retnmmg to the condition that a line shoald toucha stN'&co
we ace that if a!l four differentialsbe made to VMuehby thé
co-oidin&tesof any point, then every line through the pointmeets the surface in two coincident points; and the point ie
therefore a double point. The conditionthat a ~vea auï&ce
may have a doublepoint 19obtamcdbyeliminatingthe variables
tetween the four eqnationaT =0, &&,amdMcalled the dis-
criminant of the given qttantic (ZfMMtson J8~~a' Algebra,
194 ÛNjfERALTBttONT0FSUBPACB~
page 48). The dMemnînantbeing the remit of eMmina~onbetweenfour équations,each of thé degree M-1, containe the
coeScientaof each in the degree (M–l)', and Mthere~re ofthé degMe4 (a-1)' in thé coenicientsofthe original equation.
It!sobvi<&omwhathMbe<~6Md,th&twhen&)tat&cebas a doublepoint,the firat polar of evMy point pMseathroughthe doublepoint.
dUT~eM~MMi~WMM~by.&&,m~r happen not
merely to have pomtBin common,but to have a whole cnrvecommonto all fourmr&ces. Thts car~ will then be a douMecurveon the scrfacoC~and every point of it will be a double
point. Now we saw (Art. 283) that the Mj&ce representedby the general Cartemanequation of the M'"degree will, in
genemi,have an infinity of double tangent planes; the re-
d~MdNm~M~M~~m~~mg~MM~h~eMm&~ydFdoublepoints,whiehwill be MBgedon a certain carve. Theematencethen of these double corvée ia to be regaidod amongthe "ordinaty mngniaiitiea" of saï&cea (aee HigAer F&tMe
C%<M~,page 47).When the point aiV~'w' is a double point, !7' and AO"
vaniehidenticaRy and any linethrough thé doublepoint meetsthe saî&cein three conaecn~vepoints if it eatisnesthe equationA'P' =0, ~ndi repMBentsa coneof the seconddegree.
863. Thepdar g«<H&T<!of a jjMMtMMpoint OMa <M!~t<!6is <tcone.
The polarquadrioof the originwith regard to any snr&ce
(where,aa in Art. 241, we have introdaced<cso sa to makethe equation homogeneons)is &nnd by differentiating n- 2
times with respect to w. Dividing ont by (M-S)(a-8).8,<!hepotar<ptMMo!S
Now the originbeinga parahoKopoint, wehave seen,Art. 2M,that thé eqttaëcn!aof the form
GENERALTSEOST0F SURFACES. 1M
02
[or, in other words, ?.'==0, and M,Mof the form «~ <o,*].
ThepoIfM'qMdnethecM
But we have seen (page 40) that any equationreprésenta a
cone when it is a homogcneoasfunction of three quantities,each of thé Crst degree. The equationjust written thcreforo
représentaa conewhose vertex is the intersectionof the three
planes, H-t+2J~'+2~+~ and y. The two former
planes are tangent planes to this cone, and y the plane of
contact.
254. It followsfrom thé tast arëole that if we form thélocus of points whose polar quadrics represent a cone, th!swill moot the surface in the parabolic pointe. This locus is&and by writing down the discriminantof A*P"=0. If a,
<P' <~ïy'b, &c., dénote tho seconddiSerentia!coefficients 'y,
&c., thé d!s<a'!minamtwill be (page 41)
This denotesa sur&ee of the degree 4 (<t-2), whichwe shallcatt the Hessianof the given sor&ce. Jh thé samo mannerthen as the intersection of a plane cmrvewithita Heœ!ande-terminesthe points of inSemom,so the intersectionofa soï&cewith îts HosaMtndetermines a curve of the degree la (a–2),whichis the lecoaof pturabottopoints (seeArt. 238).
266. It followsfrom what haabeenjust provedthat throngha given pointcan be drawn 4n (n-1) (M–&)<Mt!MMaytangentplanes (Me Art. 288). For since the tangent plane passesthrough a mxedpointa îts pomt of contactMeson the polarsmf&oe,whosedegree is n -1, and the intersectionofthie sor-face with the stn'&ce and the surfacedeterminedin thelast articleas the locns of pointsofcontactofstaûonMytangentplanes,detennime4~ (st -1) (n 2) points.
Otherwisethm) the stationaïy tangent planesto thésorfaco
196 SNtEBAL TBEORT OP SURFACES.
thmugh any point are aiso siationMy tangent planes to the
tangent cone through that point, and if the cone be eat byany plane, thèse phnes meet it in the tangents at the pointsof inflexionof the section. But the number of points of in-flexionona plane curveare determinedby the formula (JB~JtefJf%MM<~)M, page 91)
It followsthen that through any point can be drawn T double
tangent planes to the surface, where T Mthe number just de-termined. It wiUboproved hereafter, that the pointeof contactof doubletangent planes lie on the intersection of the surfacewith one whose degree is (M–2) (H*–M*+?–12).
2S6. If a right line lie altogetherin a «a~Me & <otH'<M<c&Me &MMH!and ~M)~ ~antMM CM~Ce,(C~MM~~e«H~Dublin .iMs~MKtthM?Journal, VoLiv., page 2M).
Let the equation of the surface be <c~+y~=0, and let
us eeek the resntt of making a?and y=0 in the equation ofthe Hessian,soas thus to find the points where the line meets
that snxface. Now evidently`r U, dlU, ~`
allthat sm&ce. Newevidently .j-t
aH contain
a! or y as a factor, and therefore Taniahon this supposition.And if we make a='0, J=0, y='0 in the equation of the
Hessian, it becomesa perfect square (~–M~)% showingthatthe right Ime touches the Hessian. If we make a!=0, y=0
in ~–my, it reduces 1~ evident thatds d~a d~ofa
when the tangent plane touches all along any line, straightor cnrved,this line liesattogether in the Hessian. The reader
CCRVATUBE O? 80BFAOE9. 197
<Mmwn~tMswîthoutdi&<!nlty,withreg<tdtothemr&M!e<~+~.
CCRVATURN0F 8CMACM.
25! We prooeed next to inveatigatethe earvatmreat any
point on a sar&oe of the vanoua sectionswhichcan be made
by planespMmngthrough that point.In the first place let it be prenuBedthat if the equation of
a curvebe t(,+M,+M,+&o.~O, thé radmaof curvature at thé
M~~Nt~<MmeM&rdMc~m:~+~. For it will be
remembered that thé ordinary expressionfor the radius of
corvature indades only thé co-ordinatesof the point and the
vahea of thé firat and seconddi~rentia! coeScients for that
point. But if we difforentiatethe equationnotmore than twice,the terms got from dMEetentlating«“ u., &c. contain powersof a?and and will therefore vaniab for ~='0, y=0. The
valuestb&refbreof the diSBrentMdcoefficientsfor the origm are
the sameas if they were obtained&omthé equationMt+«, s' 0.It fbHowBhenco that the radins of corvatafe at thé origin
(the axes being rectangular) of y+<+8&~+<+&c.t=0 0
M n- (see <X?KtM,p. 206); or thia vatno cameasily be found
directty from the ordinary expressionfor thé radins of curva*ture (B)~C!*j?%MMC~MW,p. to8).
258. Let now the equation of a sm'&cere6!rmd to anytangent plane as plane of a!y and the correspondingnormalas axis of < be
and let usmvestigate the curvature of any normal eection,thatM of thé fteetîonby any plane paasing through the axh of e.
Thua, to find thé radius of corv&tmteof the section by thé
plane <c~we hâve only to make y=0 m the equation, andj.
we get a curve whose radius of curvature is In like
manner the section by thé plane bas its radius of curvaturo
==~.And in ordor to find tho radius of cnrvaturo of any
CUttYATURE(HPaCNFACES.198
aeettoa whosephme makes an angle 9 with the plane <M,we
bave only to tmn thé axes of <cand 11throngh an angle 6
(byeabstitatmg~cM~–ysm~fM'a~Md~ain~+ycos~&ry,<Xw«M,p. 7); and by then putting y=0 it appetM'aas beforothat the radius of curvature is ha!f the reciprooal of thé newcoefficientofa~ that is to say,
269. The reader willnot fail to observe that this expressionfor the radius of carvatnre of a normal sectton is identicalinform with the expressionfor thé square of the diameter of acentral cornein terms of the angles wbich it makes with théaxes of co-ordinates. Thus if p be the seau-d!ameter answer-
ing to an angle of the como Ac*+2~ty+ 6~='~ we have
~p'.It may be seen otherwiae that the radu of cnrvatare aro
connected with their directions in the same manner M the
squares of the diameters of a central conic. For we haveseen that the mdii of curvatnre depend only on the terms in
M,and u,. The radii of cnrvature therefore of ail the sectionsof «,+M,+M,+&c. are the same aa those of the sectioaa ofthe quadric«,+«,; and it was proved (p. ÏM) that these areaMproportionalto the squares of the dtameteN of the centralsectionparallelto the tangent plane.
It ia plain that the conic, the squares of whose radn are
proportional to the radil of carvatare) is a!mu!arto the in-dicatrix.
260. We can now at once apply to thé theory of thèseradii of curvatare all tho resolts that we have obtained for
thé diametera of central conica. Thus we know that the
quantity ~lco~6+2J3coaP9im6+ONn*6 admits of a maxi-mnm and ~!n!mnim)~a!ue; that the values of which corre-
spond to thé maxinmm amd minimum are aiways reat, and
belong to directions at right angles to each other; and thatthosevaluesof are given by the equation(aee CbMMs,p. 140)
CUBVATUBEOB'BUMAOB8. M9
Hence,atMypomtenasaï&cethe)'earoamoag~Mn<a'mat6ectioM,<me&rwhichtheTatoeofthetaditMofcarvatttreis a maximum and one for whîch it M&minimum; thé d!teo-
tions of these )!Mtiûn9are at right angles to eachother; and
<heyaM~d!recëomofthea~oftheM<~tnx. Theyplainly bisect the angles between the two Memonal tangenta.We ahaHca!l theoethe jM~c<p<tJsections,and dMcorrespond-ing radii ofcorvature the principal )wKt.
If we turn round the axes of mand y M as to coincide
with the ditecëons of maximum and minimumcorvatore j<tst
<~Mmm&hhMm~~&eqm~~y~~+&&y+C~will take thé form ~'as*+JBy. Now thé formulaof the last
article, when thé coeScte&tef a!yvaniahes,~ves the following
expressionfor any radiusof corvatureap'=~' oo~~+ B' sm'
But evidently and J!' are the valties of ~B correspondingto0'!=t0,and~=90* nenoe any radios ofcarvatoMiB ex-
pfMsedmtemM~thetwopnne~n~pa~~andafthe angle which the direction of ita plane makes with thé
principal planes, by the formula
Itia plain (as in CM~, p. 143)that and f~, or~r
atepTenbyaqMdra~oeqn&tK)n,theeamofHteMqtmnt!t!esbeing~+ 6' and their prcjact J~C'
When ~=~ <J1the other radn of carTatuMare à!so<=p.Thé &nn of thé eqaadon then ta j!!+~(~'+~')~-&o.=0, orthe imdKi&trixMa cu'ek. The origin iathen an «mMoc.
From the expreMMBOin &!a articleve deduoeat omoe,aa
in the theory of oemtMlcornes, that the «MX Mo~MW!t!t&of ~<MKt0/' CMtIXttMMC/'<tPO<MM!M!<ectM!M<!< <t~&e? eacho<~ CMM<afK</and again, <M<w<t~<ee<MH<be MM<&
<!&M~~a pair of coN~< foM~wt~(see Art. 387), the SMM
0~ their Mt<Kt< CWCNtMMMCOtMtttMt.
Thisfonauh(wtththéiofeMncmthewa&omlt)MductoEuteï.
MO CUBVATURE0F SUMACS).
Ml. It will be obsèrvedthat the radius of curvature, beingproportionalto the square of the diameter of a central oonio,does not becomeimaginary, but only changes sign, if the
quantity J[cos~~+8J&cos~s!n~+Csm'~ becomes negative.Now if radii of curvature directed on one aide of the tangentplane are consideredM positive, those tomed the other waymust be consideredM neg&~ve and the alga changes whenthe directionia changedin which the concavity of the curveMtttmed.
At an ellipticpoint on a sur&ce that ta to aay, when .B*is leu than j~C, the aSgnof cos'~+SBcoa~Nn~+Cstn'~remains the same for ail values of <?;and therefore at aucha point the concavityof every section through it ia tumed mthe samedirection.
At a hyperbolicpoint, that is to say, when F* ie greaterthan ~C, the radius of cnrvatnre tw!cechanges eign and the
concavityof some sections is turned in an opposite directionto that of others. The sar&ce in fact cuts thé tangent planein the neighhoarhoodof the point, and the innexiona! tangentsmark the directionsin vhich the surface crosses the tangentplane and dividethe sections whoso concavity !s turnod one
way fromthose whichare turned thé other vay.* And when
we have chosen a hyperbola the squaresof whose diametersare proportionalto one set of radii, then thc other set of radiiare proportionatto the squares of the diameters of the con-
jugate hyperbola.
262. Having shewn how to find the radius of ccrvature
of any normal section, we shall next show how to express,in term of this, thé radius of curvatnro of any obliquesection,inclinedat an ang!e to the normal section,but meetingthe
tangent plane in the same line. Thus we have seen that the
radius of curvaturo of the normal section made by the plane
The illustrationof the Mmndtof a monatainpasa~H eMMethéreadertoeoneeïwhowa Mt&eemayin two directionsMakMow thetangentplane,andontheotherMdesrise aboveih Theshapeofa saddleaffordsanotherfamniarillustrationofthesamething.
CORVATMB0F 8UNFACB8. 201
ye.0!a,r. NowletuatomtheMteeofyamdetmmdm
He!r plane through an angle (wMeh!s d<OMby eabstitcdBg<!coa~-y mN~fbr e Mide sin~+y coe~ &try). If we nowmake the newy ~0, we ehaUget the equation(st!Nto tect-
angniar MMa)of the eecëomby &ptaM maHag an angteimththé oMplane ~"O, but etiû paasmgthMughthe dd ams
ofa;; and th!*eqaatton will p!amtybe
and by the Mme method M before thé radiu of carvatore is
found to be 'oTt 'S '=~oos~, where jB ia the radius
of curvature of tho corresponding normal section. This iaM.EUNIER'8TBEOREM,that the radius of curvatureof an oMt~MeMc<MaM equal to the ~o~ecf&Mon the plane o/' <~MMc<~Mtoftheradius of curvatureof a normal section~MMM~atM~A the«MM<(H!~eK<7MC.Thus we aee that of aUoectioMwhiehcan
be made through any line ârawn in the tangent plane, thenormal sectionis that whose radius of carv&tureis greatest;that is to eay,the normal section!a that whichis leastcorvedand whichapproachesmost nearly to a straight!ine.
Meunier'atheorem bas been already provedin the case ofa quadric (see p. 159), and we might therefore,if we had
chosen,have dispensed with giving a now proof now; forwe have aeen that the radius of curvatureof any sectionof
«,+M,+M,+&c. is the same as that of the correspondingsectionof thé quatMo M,+M,.
263. Every sphère whosecentre is on a normalto a surface,and which passes through the point whero thé normal mcctsthe surface, of course touches the eurface. But the contactwill be of thé kind catted stationary contact (Art. t89) when
thé lengtb of the radius of the aphere ie equal to one of the
principal radii. For if the equations of two mrfaces which
touchbe
m CC&VjMfPM0FM!M'ACBS.
paeeMtbrough their carve of !n<emect!on,and it WMproved(Art. 128)that the three tenns just written represent the tan-
gente to thé caïv~ of htemeo&m of the Mt&ees. Theae
tangents coincide,or ~teM is etationary contact (Art. 189)vhen (~ ~t') (<7- 6") (~ F)'. When ~= JB' = thiaconditMnimptieseither ~1=~' or C'=C'. The surface then
f!+A)~+C~+&<=tO 0 will have etationary contact with the
sphère 2M+a~+~j!=0, if3fa'orc'
But thèse are
the vtJnea of the pnndptd tMH!.
264. The primdpleBlaid down m the !<tatarticle enableus to find an expressionfor the vaines of the principal radnat any pomt; the axes of co-ordinates having any posMon.It will be observedthat what we have proved is, that if
«,+«,+&c., «,+~+&c. fept'eaenttwo snr&ceswhich toach,then thé intersectionof thé planeu, with the cone u,-v, g!veathe two tangents to their corve of intersection and there is
stationary contactwhen the plane touches the cône.Now if we traDsfbnn the equation to any point a/y'a' on
thé satËMeas origin, it becomes
or !f we denote the first diferential eoeSdents ty Z, J~and the secondby a, e,&c. as before
The equation thon <tf any sphère having the same tM~ontDianeia
amdthe spheK will haveatattomuy contact with the quadric ifX bodeterminedso asto a&tM~the conditionthat Za:+~ -t-~&~aUtonch
CCRVATUBE<? BtJBPACM. 208
when the absolote term rednces to where JSTis the
HeaMan,written at Ml length Art. 68. We might have seena jM'tM that for any point on the Hessian,the &tMohtetermmust vanish. For since the directionsof the principalsectionsMaectthe angles betweenthe Mextonat tangents; when theintlexionaltangents coincide,one of the principalsectionscoin.cideswith their commondirection,and the radius of curvatureof this seetion is infinite, sinoethree consecutivepoints are ona right line. Hence one of the values of (which is the
reciprocal of r) must vanish. By equating to nothing thecoeNoMntof in the preceding quadratic, we obtain the
equation of a aat&ce of the degree SM-4, whidt mtemectathe given surface in tJl thé points where the principal radiiare equal and opposite: that ia to say, where the indïoatnxis an eqd!ateral hyperbola.
CURVATUIE OP SURFACES.204
Theqntdrat!coftM9M'tMenMghtaboh<tvcbeen&undat once by Att. 98, whieh ~TM the axes «f &MC~onof the
quadric
madepara1lel to thé plane Za)-tJt~+J~=Û.
265. From thé equations of the last article we caa Bnd
the radius of carvatnM of any normal aection meeting thé
tangent plane in a !me whose direction-anglcaare given.For thé centre of corvatm'e lies on the normal, and if we
deambe a sphère with tbis centre, and radiua equal to theradius of onrvatnro,!t muat touch the surface, and its équationis of the form
The consecativepoint on that section of thé sor&cewhich we
are comddermgsat!s&esthia equation, and also the equation
And smce this equation is homogenoous,we may write for
<c,y, the direction-cosinesof the line joining tho consecutive
point to the origin. As in the iMt artideA. ==~
Hence
~V(z'+~'+~')_acoa'a+~co~+cco~y+2?cos~cosv+2MCOf)~coM+Sxco8<!tCo~'
The problem to find the mamnmm and minimum radius ofcnrvatnre is therefore to mako the quantity
a maximumor muummnBuMectto the relations
And thu we see again that this is exactly tho same problemas that of &MUagthe axes of thé contrat section of a q~m~cby a planoLx + My+?.
CCBVATUM:0F SURFACES. 205
266. In like manner the problem to nnd the <ec<t!M!<ofthé pnmapat sectionsat any point !a the Mme M to find thedirectionsof the axesof the sectionby the plane2~ + Jh~ + J<%of thequadricoa~+6~'+o!'+ 3~+8NMa!+2M~'=1.
Now given any diameter of a quadnc, one section canbe drawn throngh it having that diameter for an tŒM; theother axis being plainly the intmection of the plane perpen-dicolar to the given diameter with thé plane conjugate to it.Thus !f the central qnadrtc be !7==1, and the given diameter
pas9 through a~y~, then the diametcr perpendicularaad con-
jugate is thé intersectionof the planes
If the former diameter lie in a plane ~'+~'+~ thélatter diameter traces out the cône which is represented bythe determinantobtained on eliminating a~' from the three
precediDgequations vu:.
And thie cone most evidently meet tho plane Za!+~-t~&in thé axes of tho section by that plane. Thus then thédirectionsof thé principal sectionsare detennmedMthe inter-
sectionofthe tangent planeZa! +My+ .Mswith the one
267. The methods aaed in Art. 2M enaMe us aleo easily
to &td the conditions for an <)mb!!ic.* If the plane of a~ be
We might Bnd the condition for an umMMeby forming the condition
that thé qaadtt~tic of Art. 2M thould have equal roots. But, M tt p. 9%thie quadratio having its roots always t'e<J M one of the c!M8 diMUMed,
Bi~Xtr ~~M, p. IM; whoM diMtimHMntcan be MpteMed as the
206 CPHVATUNEor KUM'ACES.
thé tangent plane at an umbilio the equation of the aurface
ia ofthe <bna
ît Mevidentlypossible so to choose X(namely, by taking it
c*~) that all the terma in the remainder shaU be divIsiMe
by <. We sec thns that if M,+M,-(-&c.represent tho surface,and M,+~e, any touching sphère, it is possible, when thé
origin is an umbilic, so to choMe that «,–Xw, may contain
u, as a factor. We see then by transformationof co-OK~nateeas in Art. 864, that any point a/y'e' will be an umbUicif itia possibleso to chooae that
Elumnatmg betweenthèse equationsweobtain for an mnMI!cthe twoconditions
Since there are only two conditionsto be eatisSed, a surfaceoftim~ degreehMmgene~adetermmatemttmbet'cfmnbiUcs; for the two conditions,each cf wMdt represen~ a
eomofsqaMM.If thereforewe onlyeoMtderreat umMMM,thetemïtof eqM<iBf[thediMtiminMtto nothingit eqaiMkntto twoconditions,whiehcanbe moreeasilyobtainedMin the text.
COBVATCBBOP SCBFACBS. 207
MM&ee,combinedwith the equation of the given sor&eo dé-termine a certain number of points. It mayhappen howeverthat the surfaces represented by the two conditions intersectin a cnrve wMchUes (either whollyor in part) on the givensat&ce. lu aucha case there would be on the given aniiaco
a Une, every pomt <~ whieh would be an umMMo. Such aline ia called a Uneof ephericalCM~M~fM.
268. There M one case in which the conditions of thelast article are not applicable m the form in wMch we havewritten them. They appear to be satisfiedby making Z==0,
~'+cj!f-3HC~0=* whence vo nught conclade that thé
eat~ce Z==0 most always pasa thronghumbilicson the givensurface. Now it t8 easy to see geometncallythat this ie not
the case, for L(or -y-)
is thé polar of the point yxte with
respect to the surface, so that if L nocessanlypassed tbroughmmhmcsit wooldfollowby transformationof co-otdma.testhat
the&'stpoIarofoM~y point passes throngh nmhiUcs. On
referring to the last article, however,it will be seen that the
investigationtacitlyassumesthat noneof the quantitiesZ,~ N
vanish; since, if so, Mme of the equationswhichwehavensed
would contain inanité tenns. Snpposingthen L to vanish,we must examine directiy thé condition that J~+~& mayhea&ctorin
We must evidently have X<=o,and it is then eamiyseen that
~'+cM'-8MOirwe mast, M:beMFe,haveo=- ~whue
m
addition mmeethe terme aM<M!+2<M!ymnet be divisible by~+~&,wemtMtha.ve~!&t*=jMt. Combiningthen with thetwo conditMMNhere found, Z<=0, and the eqatttMn of the
snt&ce, there are four oondMomwhich, exoept in spedM
cases, cannot be ea~s&adby the co-ordinatesof any pomts.If we clear of &actiona the conditionsgiven in the last
article, it wiU bo found that they each contain either L, ~M,or jV as a factor. And what we have proved in thie article
M8 COWATUM! Or 8CNFACE6.
is that thèse &ctors may be snppremedas irrelevant to thé
questionof mnMMca.*We now proceed to draw aomeother in&rencesfrom what
was provcd (Art.263) namely, that the two principal ephereahavestationarycontactwith thc snr&ce.
<
269. tFXeKtwo aM~MMhave<<a<t<M<M~contact,.~<y <o«c~<? <MOC<MMeoM<M)epoints.
Theequationacf the two sar&cesbeing written as mArt. 263,thé tangent planesat a consecutivepoint are (Art. 287)
That thcsemay be Identioal,wemusthave
whieh is the condition for stationary contact.
The sphère, therefore, whose radius is eqnal to one of the
principal radii touches the snr&cea in two consecutive pomta;
Erom what bu been Nud we caminfer the nmnber of ambSies whioha surface of the ft"*degree will in general poMeM. We have seen thatthe umMica are determlned as the intersection of the given surtaoe with
a curve whose eq~Mt!oMaM of the form'y 'M °7<t
Now if ~i, B, C
te of the degtee t md d', B', C' of the degree M, then ~JB'
~C'- CM*are each of the degtee t+M, and interMot in a curve of the
degree (<-Hn)*. Bat the mteKection of theee two «.tr&cea imdudes thecurve dd' of thé degree ~t which does not lie on the surface BC'- CB*.The degree therefore of the eurve in question is <*+ &Mm*. In thé
prêtent tMe ~-Stt-4, <M°2~-a md the degree of the curve wonidaeem to be 19«* 46a + 28. But we have aeen that thé ayatem we are
diMUMin~ineludea three Ottnret such as
whieh do not pMathrough umbilics. Subtmcdngthereiore from thenumberjoat foud 3(M-1) (3<t 4), we see thtt the ambilicaare detet-mined asthe iatetMttxmof the given Mf&eewitha curveof the degree(10~ ?? +16),and thete&Mthat the numberof mabiMcsh in generalK(10tt'-M<t+M).
CHRVATURE0F SURFACES. 209
p
or two consecutivenormals to the surfaceare aise normalsto
the aphere,and consequentlyinternectin Ita centre. Now wc
knowthat in plane curves the centre of the cirele of curvature
maybe regarded as the intersectionof two conmoutivenormals
to the curve. In surfaces tho normal at any point will not
meet the normal at a conaecutivepoint taken arbitrarily. Bntwe aee hère that if the ctmsecotivepoint be taken in the
A!rect!omof either of thé princ!pal Mct!on~thé two consécutive
nomMbwill interseet, and their commonlength will be the
correspondingprincipal radius. On accountof tbe importanceof thistheoremwe give a directinvestigationof it.
270. 2b ~t~ Mt !0%e<<!<M«<~ normal <t<any ~<M'M<OKa
~M~M&intersectedby<!consecutiveKOfma~Take tho tangent
planefor the planeof a~, and let the equationof the surfacebo
Thé <t!rect!ontherefore of a consecutive point whose normal
meets the given normal is detennined by the equation
But thMis the same equation (Art. 260) whieh determines thé
directions of maximum and minimum eurvature. At any pointon a surface therefore there are two dirocttom),at right anglesto each other, auch that the normal at a consecutive pointtaken on either, intersects the original normal. And these
directions are those of the two principal MctioM at the pomt.
Taking for greater simplicity the directions of the principalaecttOtMas axes of co-ordmates; that ia to say, making J3=0
210 CORVATURE0F SURFACES.
in the precedingequations,the eqnation of a consecutivenormal
becomes––i=S–~e=8~ whence it la easy to see that théAd a G,y,
=28, w ence it is easy to see that the
normalacorrespondingto the pointa ~=0, a!'s=0 intersect theaxis of e at distancesrespectively <==~, <=='~C. The inter-
cepts thereforeon a normal by the two consecutiveones whieh
intenect it are equal to the principal radii.*
271. We may abo arrive at thé aameconclusionsby seek-
ing the locas of pointaon a sar&ce, the normals at wMchmeeta mxednormal which we take for axis of Making <e=!0,~=0in the equation of any other normal we see that the
point where it meets the sur&ce must satisfy thé condition
.~=:The cnrve where this ani&ce meets the given
<&cear&ce has the extremity of the given normal for a double
point, the two tangents to which are the two principal tangentsto the snr&ceat that point.
The apecial case where the nxed normal is one at ancmbilicdeservesnotice. The equation of the aarCMebeing of
the formiB-)-~(a!*+~)+&c.=0, the lowestterms in the equa-
tion a!=~-?"t when we make <!=0, will be of the third<~
=OKC
degree, and the umbilic is a triple point on the curve locus.
Thus whileeverynormal immediatelyconsecutiveto the normal
M.Bertrand,inM<them of the curvatureof Mt&cM,caloulateathe anglemadebythe consecutivenormalwiththeplanecontainingtheoriginalnormalandtheconsecutivepointa'V. Supposingsti)lthed!tee-tionsof the principalsectionsto be axesof co-ordinatee,thedirection-eomMof the consecutivenormalareproportionalto &<<.)/,2<y, whilethoMofatan~enttmepetpendMalaTtothei'adiMteetoraMpMportionatto
y't ~t 0. Henoethe eo~neof the anglebetweentheaetwolines,ortheaineofthe anglewhiehthe consecutivenormalmakeswiththenormaleeetion,i<proportionalto (C-~).tV} cr, if « be theanglewhiehthedirectionoftheMMecaiivepointmakeswithoneoftheprincipaltangente,ispNpoftionalto (C-~jt)Mn2«.Whena c 0 m' a 90°,thisanglevanKheaand theconeecntivenormalis in the planeofthe originalnotmat.
CUttYATUBE OP SURFACES. su
P2
at the umbilie meets the latter nonnal, there are threo directions
along any of which the next following normal will aiso meet
the normal at the umbilic.
272. A ~'Meof e«!'M<'<e* on a surface is a line traced on
it Bach that thé nonnats at any two consécutive points of it
intersect. Thus starting with any point ~f on a surface, we
may go on to either of the two consecutive points j~ N' whose
normats were proved to intorsect tho normal at M. The normal
at again, M intersected by the consecutive normals at two
points P, tho element NP being a continuation of the
element J~ while the element JMP' is approximately per-
pendicular to it. In like manner wc might pass from the pointP to another consecutive point Q and so have a line of cnrva-
ture 3fMP~. But we might evidently have pursued the same
process had we started in thé direction MN'. Hence, at a&y
point Jtf on a surface can be drawn two linea of eurvature;these eut at right angles and arc touched by the two "pnn-
cipal tangents" at 3~ A line of curvature wiU ordinarily not
be a plane curve, and even in the special case where it is planeit need not coincide with a principal section at though it
must touch such a section. For thé principal section must
be normal to the surface, and the line of curvature may be
oblique.
A very good illustration of lines of curvature is anorded
by thé case of the sar&cea generated by the revolution of any
plane curve round an axis in its plane. At any point P of
such a surface one line of corvatare is thé plane section passing
through P and through thé axis, or, in other words, is the
generating curve which passes throngh P. For all the normala
to this curve are abo normals to the surface, and being in
one plane, they intersect. The corresponding principal radius
at P is evidently the radius of curvature of the plane section
at the same point. The other line of curvature at P is the
The whole theory of HnMof eurvature, ambttiea, &t. is due to Monge.See hn «AppUeation de t'Ana~'se A la CMom~trie,"p. 124, LicaviUe's
Edition.
CURYATUBJS0F SURFACES.212
cMe which is the section made by a plane drawn throughP perpendioularto thé axis of the Mr&ce; for thé normalsat all the points of this section evidently intersect the axisof the surfaceat tho same point, and thereforemtersect each
otber. The mtefoept on thé normal between P aad the axis
is plainly the second principal radius of the surface.The generating curve which passes through P is a prin-
cipal section of the surface, since it contains the normal and
touches a Uneof curvature but the section perpendicular to
the axis is mota principal sectionbecause it doesnot containthe normal at P. The second principal section at that pointwoold be the plane section drawn through the normal at Pand throagh the tangent to the circle described by P. The
example chosenserves aiso to illustrate Mennier's theorem;for the radins of the circle describedby P (which,as wehave
seen, !s an oblique sec~on of tho surface) is thé projectiononthat plane of the intercept on the normal betweenP and the
axis, and we have just proved that this intercept is the radiusof curvatnreof thecorrespondingnormal section.
278. It was proved (Art. 266)that the direction-cosinesofthe tangentUneto a principal sectionfulfilthe relation
Now tho tangent line to a principal section is also the tangentto the line of cnrvatnre; while, if <&be the element of thearc of any onrve, the projectionsof that element npon the
three axes being <&c,dy, <&,it is evident that the coeinesof
the angles which <&makes with thé axesare -j-,
The dMbrent!alequation of the Une of curvatnre is therefore
got by writing<&B,< ds for coea,cos~ co6'yin the precedingMnmuat
ThM equation may also be 6)tmd directiy as follows(see
Gïegoty's &)?' 6~MKe<fy,p. 256) Let et, ~8,'y be the co-ordmatea of a point oommon to two consecutivenormals.
CCRVATCBE0F SURFACES. 2M
Thon, if a~ be thé point where thé &~t normal meeta
the BNiAce, by the equationo of thé normal, we have
*e,<°~=~ or if we call the commonvalue of
these &actions we have
But if the second normal meet the surface in a pomt<B+<&
y+< ~+<&, then expreasmgthat e<y ft&tMesthe equationsof the secondnormal, we get the Nune reMitaM if we di~-
rentiate the precedingequations,considering<~ M conBttmt,or
<&?+Z<~+MZ<=0, <~+~<~+~f=0, <&<~+~<0,
from which equattona f)!m!n&t!"g dd, we have thé Mmedeterminant aa in Art. 866, viz.
EjMwmgas we do that the lines of curvatnre are the inter-sections of the dlîpsoîd with a aystem of concentncquadrles
(Art. 206), it wouldbe easy to aeNimefor the integral of this
equation ~a~+~*+(&='0, and to détermine the constants
by actual substitution. If we Msmnenothing as to the formof the integral we can eliminatea and da by the holp of the
equation of the surface, and M get a dMerentia!equation in
two variableswhichis the equationof thé projectionof the Unes
214 CURVATUHEOF SURFACES.
of curvatnreon the plane of xy. Thue, in the present case,
multiplyingby and redudag by the equation of the eU!pso!d
and its dt&reNtta!,we have
or the limesof curvature are projacted on the principalplaneinto a aenes of comca whoBeaxes <t',&'are connectedby therelation
It ia not dKEcntt to see that this coincides with thé account
given of the lines of curvature in Art. 206.
274. The theorem tbat confocal quadrics intersect in lines
of curvature is a particular case of a theorem due to Dnpin,*whieh we shall state as followa If three ~!<MM MtMtect at
rigAt angles, and < each pair a&o tM<g!'«c<at right anglea at
<%eM'next <!<MMMM<tMC<MKMM<point, then <Ae<?:')'ec.tMM~of the
M!<ef<ec(M<Mare <Xedirections of the lines o/'eMfcatttfcon each.
Take the point common to ail three surfaces as origin, and
'~e titrée rectangular tangent planes as co-ordinate planes; then
the equations of thé surfaces are of tho form
Dêvdoppements de Géométrie, cinqai&meM~noîre. Thé demoMtta-
tion here givenle by PM<!eMorW. Thomson: see Gregory't t8MMCMm<<ry,
p. 263. Otm~-t%<JM<~«<M««M~<KM<nM<,Vot. tv., p. 82.
CUBVATPBE0F SURFACES. 21&
At a consecutivepoint commonto the Smtand secondsurfaces,we most hâve a:==0,y='0, <='~ wherea' N very ama! Theconsecutivetangent planesare
Fonnmg the conditionthat these ehoatdbe at right angtesand
onlyattending to the terniswhore is of the 6tst degree, we
have6+y='0.In like manner, in order that the other pairs of sar&cea
may eut at right angles at a consecutivepoint, we must have
~'+y=0, &"+&=0, and thé three equationscannot be ful-
filled nnleaa we have &,& &"each eeparately=0; in whichcaMthe form of thé eqnationashows(Art. 260)that the axesare the dtKCttonsof thé lines of curvature on each. Hencefollowsthe theorem in the form given by Dapm namely, that
~(~e lie<~Mes~s<e!!Mq/'ott~Mes,suchthat every~Mr~ceof one
~<<em~cut at right <Mt~Mbyall «~ surfacesof tlteo~e~ t!oo
s~teMs,thenthetMfe)'<ec<t<Mof tm surfaceste!M~Mytod'~reKtey<<eMS& a line of c<<MM<tt<'e<Mteach. For, at each point of
!t, it !a/by hypothesis,possibleto drawa third snr&ceeuttingboth at right angles.
275. If two M<)~)!!CMoutat right <t)~r~* and if f~' w<e~-sectionMa line of cMM!a<«MOKone, t!fis o&oa line o~curvatureontheother.
Proceeding as in the taat article, and taking the origin at
any point of their intersection,wemust, in order that they mayeut at right angles, have5 +&':=0,whenceif &==0, &'=0.
Otherwisethus the direction-cosinesof the tangent planesof the two eorfacesbeingproportionalto Zf,JM,N; L', jM',JV*the direction-oosinesof their lïno of intersection are propor-tional to J)CV-3f'~ ~L'z, Zjtf'Z'if; and in orderthat thia intersection shouldbo the directionof a line of cnrva-
Thisiaalsotïueiftheyeutnt anyconstantangle.
CUBVATUBËOF SURFACES.216
tore on thé ~Kt surface, we must have the coadttion foMM
(Art. 278)
but this is the conditionthat the Uneof intersection shonidbea line of curvatoreon the secondsurface.
276. A Une of curvature is, by definition, auch thatthe normals to the surface at two consecutivepoints of itintersect each other. If then we consider the surface gene-rated by aU the normals along a line of curvature, thia will bea developablesurface (Note, p. 75) stnco two consecutivegene-
rating Unesintersect. The developablegenerated by the nor-
malsalonga line of curvature manifestlycuts the given surface$t right angles.
The locas of points where two consecutivegeneratore ofa developableintersectis a curvewhose propertieswill be more
fully expkmed in the next chapter, and whieh is called the
C!Mp«Medgeof that developable. Each generator is a tan-
gent to this carve, for it joins two coMecutivepoints of the
cnrve; namely, the points where the generator in questionia met by thé preceding and by the eacceedmggenerator (seeArt. H9).
Consider now thé normal at any point Jtf of a stM&oe;through that point can be drawn two Unes of corvatate
CURVATCBE0F SNBfACEf). 217
JMKP~, &c., JMyjP' &c.: let thé nomals at the pointaJM;N, Jf~ &c. intenect in C, D, JE,&c., and those at
JM,JV',JP*, in C', D*,JF'; then it is evident that thé curve
CPJS,&o. is the caspidaledge of the developablegeneratedbythe normals along thé first Uneof curvatnrewhile C'D'JS"iathecnspidal edge of the developablegeneratedby the normals
abng the second. The normal at J~ as haa just been ex-
plained, touches these curves at thé points C, 0' which arethe two centres of carvature correspondingto the point M.
What haa been proved may be stated as fbtiows: The
caapidd edge of the developablegeneratedby thé normala
along a line of curvature, ia the locusof one of the syatemaof centres of curvature correspondingto all thé pointaof thatUne.
277. The aasemMageof the centreaof curvature C, C'
answering to all the points of a surface is a surface of twoaheetscaUedthe ~Mj~tceof centres(seeArt. 208). The cnrveODE liea on one sheet while C'JO'JB'lieson thé other aheet.
Every normal to thé given surfacetouchesboth sheets of thesurfaceof centres for it haa beenprovedthat the normalatjf touches the two curves CDE, C'D'JF',and every tangentIme to a curve traced on a surface is abo a tangent to theaudace.
Now if from a point, not on a Barfaeo,be drawn two con-aeoitive tangent lines to a surface,the plane of those lines ia
manifestlya tangent plane to the surface;for it is a tangent
plane to thé cone whiohis drawn from thé point touchingthe
surface. But if two consécutivetangent Knesintersect on the
surface, it cannot be inferred that their plane touches thesurface. For if we eut the surfaceby any plane whatever,
any two consécutivetangents to the curve of section (which,of course, are also tangent lines to the surface)intemectonthe
carve, and yet the plane of thèse Unesis Mpposednot to toueh
thé surface.
(Jonaider now the two consécutivenormalaat the points
JM,J~ these are' both tangents to both aheets of thé surface
of centres. And since the point C in whiehthey intersectis on
CCRVATCBBor BOBFACES.218
the amt sheet but not BecemafHyon the aeeond,the plane ofthe two normals is the tangent plane to the second aheet of
thesm&eeofceotMs.The plane of the nonnata at the pointe JK,N' is the tangent
plane to the other sheet of the soï&coof centres. But hecaasethé two lines of curvatnre through M are at right angles toeach other, it followsthat those two planes are at right anglesio eachother. Hence~the <<M~eat~&MMw<cthe«o~aceof oentreaat the twojpMM~<~<?', M~e CHynormal Mee<<t!f, eut eachc<~)' at y~~ <a~
It is manifestthat for every umbilic on the given surface,the two sheets of the surface of centres have a point commonor, in other words, the surface of centres bas a double point;and if the original surface have a Une of spherical curvature,the surfaceof centres will have a double line. The two sheets
willcet at right angles every wherealong this doubleline.
878. It ia convement to definehere a ~eod5MM&'Mon a
sorface, and to estaMIsh the fondamental property of sucha line; namely, that tts oscnlatmgplane (see Art. 119) at anypoint is normal to the surface. A geodesic line is the formaœumed by a atramed thread lying on a surface and joining
any two points on tbe surface. It is plain that the géodésieis ordinarilythe shorteat line on the surface by which thé two
points can be joined, mnce, by pnUIng at the ends of the
thread, we must shorten it as much as the interpositionof the
surface will permit. Now the resultant of thé tensions alongtwo consecutiveelementaof the curve, formed by the thread,lies in the plane of those elements,and since it muât be de-
atMyed by the resistance of the surface, it is normal to the
surface; hence,Meplane of <MO<MMecM«Me&MeH<eof Me~e~-<&McccM&!aMtha normal <o the MH~ce.*
1have followedMonge in giving thu proof, the mechanical principleswhieh it involves being so elementary t)Mtit seeaM pedantio to object to
the infodnction of them. For the bene&t of those who would pte&t a
purely geometrical proof 1add one or two in thé text. For Mttdere&nt!!iM
with the theory of maxima and minima it MMMeetyneceMary to add that
OBttVATME0F SURFACES. 21&
The Mme thing may abo be provedgeometnca!!y. In théfirat place, if two pointa~i, C in dirent planes be eonnected
by joining each to a point B in thé mtemecHonof the two
ptanes, the sum of AB aad BC will be lesa than the snm of
any other joining lines .B', J~'C, if ~& and BC make equalangles with T?", thé mtemectionof the planes. For if one
plane be made to revolve about TT until it coincidewith thé
other,AB and BC becomeone right line since the angle TBAts anpposed to be equal to fJSC; Md the right line ~4<7!sthe shorteat by which thé points A and C can he joined.
It followsthen that if J~B and BC be consecutiveelementsof a curve traced on a surface, that curve will be the shortestline connecting A and 0 when AB and BC make equalangles with J?~ the intersectionof the tangent planée atand C.
We see then that -<tB (or its production)and BC are con-sécutive edges of a right conehavmg JM*for ita axia. Nowthe plane containing two consecutiveedges is a tangent planeto the cone; and since every tangent plane to a right côneis perpendicnlar to the plane containingthe axis and the lineof contact, it followsthat the plane ABC (the oscolatingplaneto the geodeHc) is porpend!cnlarto thé plane AB, By whichis the tangent plane at The theoremof this article is thusestaHished.
M. Bertrand haa remarked (Z~'oMOtNe,t. xni., p. ?3, cited
by Cayloy, QuarterlyJournal, VoLt., p. 186) that this funda-mental propcrty of geodemcafollowsat once from Meunier'atheorem (sec Art. 262). For it is evident, that for an inde-
nnitely small arc the chord of which !a given, the excess in
length over the chord is eo much thé leas as the radius of
curvatnre is greater. The shortest arc there&re joining two
a geodeMcneed not be the abMhttety shortest line by whiehtwo points on
the surface may be joiasd. ThM, if we conaidertwo points on a sphere
joined by a great circle, the remaMn~ portion of that great circle, CMeed-
ing 1800Ma géodésie though not the shortest Une connecting the point!.The geodeaio however will alway<be the shortest line if the two points oon-
aidered be taken satncieot!)- near.
CURTATCBE0F SURFACES.MO
inde&tMy near pointa J?, on a surface M that which basthé greatest radins of cnrvature, and we have seen that tbisis the normal section.
279. Returning now to the surface of centres, 1 say thatthe curve ODE (Art. 277)whieh !s the locusof pointeof !ntep.section cf consecutivenormala along a line of curvature Ma geotksMon the aheet of the surface of centres on whichitlies. For we aaw(Art. 277) that the plane of two consecutive
normalato the surface (that H to say, the plane of two con-
secntive tangents to this carve) ia the tangent plane to thesecondsheet of the surface of centres and !s perpendiculartothe tangent plane at C to that aheetof the surface of contres
on whichC liea Since then the oscolatingplane of the cnrve
ODE Malways normal to the surface of centrea,the curve is
a geodesioon that sm'&ce.
280. We have given the equations connectedwith Unesofeurvatureon the suppositionthat the equation of the surfacehas been given, as it ordinarUyis, in the form ~(<c,y,<!)~0.Aa it ia convenient,however, that the reader dtoald be able
to findhere the fbrmulœ wh!chhave been commonlyemployed,we shallconclndethis chapter by giving thé principaléquationsin the fom given by Monge and hy most subséquentwriters,viz.whenthe equationof the surfaceis in the form e == (.c,~).We usethe ordinarynotations
<b*=~<&!+~y, <~=f<&!+~, <=s<&!+<t~.
We might derive the reaults in this form from those found
ah'eady; for since we have !7'='~(a',y)–~='0, we have
with correapondingexpressions for their second dîSërentM
coeScients. We ahaU, however, repeat the investigationsfor
this form as they M'amsuaUygiven.The equationofa tangent plane is
MlCURTATURE0F 8UBFACEB.
This eqnationdéterminesthé projectionson the planeof xy ofthe two directionsin which consecutivenonnàb can be drawn80 as to intersect the given normal.
281. From the oqu&tîoMof the precedingarticle we eau
also find the lengths of the principal r&dS. The equations
<&?+jp<&!=('y-<s)< <~+~'=(<y-e)<
when transformedas abovebecome
CUBVATME 0F SUKFACE8.222X<M
Et!mn)Kti!)gthen y-ar by the help of thé last eq<M<ion,JS H
given by the équation
282. From the preceding theorems can be dedacedJoMbimsthaI'stheorem (see C'fe&,Vol. XXX.,p. 847)that if a
I!ne of curvatnre be a plane curve, its plane makes a constant
angle with the taagent plane to the surface at any of the
points where it meets it. Let the plane be e=0, then the
equation of Art. 278
becomes<&:<t~< But we have alsa ~.c't-go~s'O, con-
seqttenHyjK~-)- 0 + g* constant. But p' +q' Mthe
square of thé tangent of the angle wMch the tangent plane
mates with the plane icw: since cos'y=-Tr-s–r.V(l+F*-)-f)
Otherwisethus (see HonvIUe, Vol. XL,p. 87) Let JMf',~tf~tf' be two consecutive and equal éléments of a line of
curvature, then the two consecutivenormals are two perpen-dicatarato these lines passing through their middle points jf,and C the point of meeting of the normalsis equidistant from
thé Unes.Mf, 3f~ But if from C we let Ml a perpen-dieutar CO on the plane .~CMW', 0 will be atso equidistantfromthe same éléments; and therefore the angle CyO==(XfO.It is provedthen that the inclination of thé normal to the planeof the line of cnrvatare rcmams unchanged as we pass from
point to point of that Ime.Moregenerallylet tbe line of curvaturenot be plane. Then
aa beforethe tangent planes through JtMf and through~f~f'make equal angles with the plane 3fM*Jtf". And evidentlythe angle whieh the second tangent planemakeawith a second
osculatmg plane Jtf~f'~f'" differs from the angle wh!ch it
makes with the nrst by the angle betweenthe two oscnlatmgplanes. Thns we have Lancret's theorem, that along o line
O/'CMtTX~MMtAe<:<!)T<t<tOHin <~ angle ~<MMMthe <<tK~e!ttplaneto <~e~M<Ceand </5e<MCM&t<M!~plane to curve Mequal <~<~eangle&e<MMMthe<tpooecaXt<M:~F~a<Mi).
228CCBVATURBOP 8URFACE8.
For exemple, if a line of ct~~atMw be a ~eo~Mt'c<<must
&eplane. For then the angle between the t&ngent plane and
oseulating plane does not vary, being aiways right: therefore
the oseulating plane itself doesnot vary. From the Mnne prin-
eiples we obtain a simple proof of the theorem of Art. 275.
283. FinaUy, to obtain the radius of corvature of anynormal section. Since the centre of curvature a~*y Ues on
thé normal, we have
AmdBineethis relationholdsfor three consécutivepoints of thesection which is oaculatedby the circle we are considering,we have
( 224 )
CHAPTER XI.
CURVESANDDEVELOFABLE8.
SECTION1. PBOJECTIVEMOPEBTtES.
284. IT waa proved (p. 13) that two eqaatMns representa carve in space. Thus thé eqo&tloBSP= 0, ~= 0 reprosentthe curve of intersection of the sur&ces V.
The degree of a corve in apace ia measuredby the number
of pointsin which it is met by any plane. Thus, if C~Y boof thé M**and n" degrees respectively, the aur&ceswhich they
representare met by any plane in curves of the same degrees,whichmtersect in M, ? points. The curve PT' is therefore of
thewtM~degree.By eliminating the variables alternately between the two
given equations,we obtain three equations
whichare the equations of the pt~eetions of the carve on the
thKeco-ordmate plmes. Any one 'of the equations takenseparatelyrepresents thé cylinder whose edges are p&ra!Ieltoone of thé axes, and whichpassestbrough thé carve (Art. 24).The theory of elimination ahowsthat the equation ~(y, ~)=00obtained by etimmat!ag x between the given equations is ofthe MM"'degree. And it ia also geometrically evident that
any cône or cyHnder* standing on a curve of thc degreeMof the degree. For !f we draw any plane through thévertexof thécone [or paralld to the generators of thé cylmder]thie plane meetsthe cône mf tmes; namely, thé lines joiningthe vertexto thé points wherethe plane meetsthe carve.
A eytmdafMplainlythe limitingceaeof a cone,wheMvertexitat ia&ttty.
FK(MECT!VEPROPERTtËB0F CUBVE8. 225
Q
28o. Now, conversely,if we are given any curve in spaceand desire to reprcsent it by equations,weneed only take thethree plane curves whieh are the projectionsof the curve on
tho three co-o~dinateplanes; then any two of the equations
~(y,<!)=0, ~'(<a;)=!0, x(~y)'=~ represent the givencnrve. Bat ordinarily these w!!Inot formthe simplestsystemof equations by which the curve can be represented. For ifr be the degree of the curve, these cylindersbeing each of
the f"' degreo, any two intersect in a curveof r' degree thatis to say, not merely in the curve we are consideringbut inan extraneous curve of the degree y'–)'. And if we wishnot merely to obtain a system of equations satisned by tho
points of the given curve, but a!so to excludeaU extraneous
points,we must preserve the system of three projections; forthe prqecdon on the third plane of thé extraneous curve inwhieh the first two cylinders intersect will be different &omthe projection of the given curve.
It may be possibleby combiningthe eqnationsof the three
projectionsto arrive at two équations&==0, 1T=:0, which shallbe satisnedfor the points of the given corve, and for no other.Bat it is not generally true that eMry curve in space is the
complète intersection of two surfaces. To take the simplestexample, consider two quadrics having a right Unecommon,as, for example, two cônes having a commonedge. Theintersection of these surfaces,which is in general of thé fourth
degree, must consist of the oommonright Une,and of a cnrveof the' third degreo. Now since the only factors of 3 are 1and 8, a cnrve of the third degree cannot be the complèteintersection of two surfacesnnleas it be a plane cnrve butthe curve we are consideringcannot be a plane cnrve,* forif so any arbitrary line in its plane would meet it in three
points,but snch a line could not meet either quadrio in more
Can-esm spacewhiehare not planecanreshavecemmoalybeencaUed"cunea of doubleeaMatm'e."in what~Bowa,1 usethe word
cuïvx"todénotea eurvein space,whichordinarUyMnota planecurve,and1 add the adjective"twiated"when1 wantto ttate exptesdythetthe curveia Mta planecurve.
MtOJEO-ÏYEPBOCEBTIESOPCUBVfN.226
than two, and therefore could not pass through three pointsof their curve of intemeotton.
886. If a curve be either the complète or partial inter.
section of two surfacesU, F, the tangent to the curve at anypoint i8 evidently the intersection of the tangent planesto thetwo surfaces,and is representedby the equations
The direction-cosineaof the tangent are plainly proportionatto JCV'-Jtf'~ NL' N'L, Z~Z'J~ where Z,~&c. are
the &Kt differential eoe~cicnts.An exceptional case anses when the two sm~acestouch, in
which caae the point of contact is a double point on their
corve of intersection. AU thia has been explainedbefore (seeArt. 128). As a particular case of the above, the projectionof thé tangent !ine to any curve !s the tangent to its projec-tion and when the curve is given as the intersectionof the
two cylinders y=~Mt a!==~'(<)) thé equations of the tan-
gent are o.
ThiB may be otherwîae expressed aa Mowa: CoMKkranyelem<!ntof the curvo <&; it is projected on the axes of co-
orduMtteainto <&),< <&. The d!rectMm-cos!nesof thisélément
are thereforeT 's
and the équationsof the tangent are
Since the eam of the squares of the three cosinesare equal to
mti~, we bave <&=.«5E'+<t<We ahaHpostponeto another section the theory of nonntJtL
KnHlof corvature, amdin ahort everything which involvoathe
227PBMECnVE PROPEBTtM 0F CUBVM.
Qi
considerationof angles, and in this section we shall onlyconsider what may be called the projective properties of
curves.
287. The theory of curves is in a great tneaMre identical
with that of developableson wMchaccoant it is necessary toenter more fully into thé latter theory. In fact it was proved
(Art. 119) that the reciprocatof a séries of points forming a
curveis a sériesof planes envelopinga developable. We there
ehoweAthat the points of a carvo regarded as a system of
points 1,2, 3, &e. give rise to a systemof lines; namely, thelines 18,28, 84, &e. joining each point to its next eonMcnttYe,these linesbeingthe tangenta to the curve and that they àbo
give r!se to a system of planes,vis. the planes 128, 284, &c.
containingevery three consecutivepoints of the system,thèse
planes being thé osculating planes of the cnrve. The as-
semblageof the lines of the system fbrms a surface whose
equationcanbe found when the equation of the curve is given.For the two equationsof the tangent Ime to thé curve involve
the threo co-ordinates y', wMchbeing connectedby tworelations are reducible to a single parameter; and by theéliminationof this parameter fromthe twoéquations,we obtainthe equation of the surface. Or, in other words, we must
eliminate a! between the two equationsof the tangent andthe two equationaof the curve. We hâve sa!d (Art. il 9)that the surfacegenerated by thé tangents Is a developablesince every two consecutivepositionsof the generating lineintersecteach other. The name given to this kind of surfaceis derived from the property that it can be unfoldedinto a
plane wtthout crnmpEngor tearing. Thus imagine any serieaof Unes~< Bb, C~ ZM,&o. (which for the momentwe takeat a nmte distancefrom each other) and such that each inter-sects the consecutivein the points a, b, c, &o. and supposea surface to be made up of the <aces~<tJ9,J9&C,C!t:D,&o.,then it is evident that snch a surfacecould be developedintoa plane by tuming the face~taB round aB as a hinge untilit formed a continuationof .B6<7;by turning the two, whichwe had thns made into one face, round cC until they formed
PROJECHVE PROPEBTtKS UF CUBVE8.228
a continuationof thé aext face and so on. In the limit whenthe linesAa, Bb, &c. are indefinitelynear, the asaembhtgeof
plane elementsforma a developablewhich, as just explained,can be unfoldedinto one plane.
The reader will find no di~calty in conceîving this from
thé examplesof developableswith whieh ho is most &m)I!ar,vis. a cone or a cylinder. There is no dMealty in foldinga sheet of paper into thé form of either surface and in un-
folding it again into a plane. But it will easily he aeen to
be impoMlHeto folda sheet of paper into the form of a ephere(whiehMnot a developable surface); or, conversely,if we euta sphère in two tt is impossibleto mako the portionsof the
Mr&celie Bmoothin one plane.
288. The plane AaB containing two consecutive gene-rating lines is evidently, in thé limit, a tangent plane to thé
developable. It is plain that we might consider thé surfaceas generated by thé motion of thé plane ~o~ accordingto
someaB9tgnedlaw, the envelopeof this plane in all its positionsbeing the developable. Now if we consider the developablegeneratedby the tangent Unesof a curve in apace,the equa-tions of the tangent at any point a:'y< are plainly fonctionsof thoseco-ordinatea,and the equation of the plane containing
any tangent and thé next consecutive (in other words, the
equation of the osculating plane at any point a:y<) is alsoa fonctionof these co-ordinates. But smce a: are connected
by two relations,namely, the equations of thé curve we caneliminate any two of them, and so arrive at this MMUt,that« developableM the envelopeof a plane <oA<MMequationcontainsa singleMMaMeparameter. To make this statement botternndemtoodwe shall point out an important differencebetweenthe caseswhena plane curve is considered as the envelopeofa moveableline, and whena surfacein general îs consideredasthe envelopeof a moveable plane.
289. The eqoatton of thé tangent to a plane curve is afonctionof the co-ordinatesof thé point of contact; and thesetwo co-ordinatesbeing connectedby the equation of thé curve,
PMJfECnVEPROPEMtES0F CUBVE8. 229~MW
we caa either eliminate one of them, or etse express both interme of a third variable so M to obtain thé equation of the
tangent as a function of a single variable parameter. Theconverse problem to obtain the envelopeof a right line whose
equation includee a variable parameter bas been diacuased,JE% P&ïMCtttft~M,p. 98. Let the equation of any tan-
gent line be « =0, where u ia of the first degree in a: and y,and the conetants are 6mctiona of a parameter a. Thenthe line answering to the vaine of the parameter a +ie
u+duA
+ d'uh°
& and tho point of intersectionof these1 ~T*'t9 +~c' thé pomt of inteMect!onof theM
tli'. bhdu
Ad'&ctwo Mneais givenby thé équationsu =0, +-y-~-t &c.=
0.
And, in the limit, the point of intersectionof a line with thenext consecutive(or, in other words,thé point of contact of
any line with its envelope) Mgiven by the equationa«=0,
=0. If from these two equationswe eliminatea we obtaindathelocusof the points of intersectionof each line of the system
with the next consecutive; that is to say, the equation of the
envelopeof aU these Unes. It is easy to prove that thé resultof this eliminationrepresentsa curve to which Mis a tangent.For if in « we replace a by ita value, in terma of x and y,
~M <?t< /C?M\ <?M~C[denvedfrom the equation y
=0, we havey =* ( ) + 'y 'y
j <~M /<?tt\.<&<<&! /M'\and ~utdun
duj dat wheretd, LI, 1 id~u~
are the diff~-~)+~ (E)' dy are the
rentiab of « on thé suppositionthat etis constant. And since<~M <&<<?t<
h b,0 it is évident that are thé same as on thé sup-
position that a is constant. It followsthat the climinant in
question denotes a carve touchedby «.If it be required to draw a tangent to this curve through
any point, we have only to substitutethé co-ordinatesof that
point in the equation M'=0, and déterminea so as to satisfythat equation. This problem will have a definite number of
solutions,and the number will plainlybe the number of tan-
gents which can be drawn to the curve from an arbitrary
PRMEOMVE PROPBB'HM 0F CU&VE8.MO
point; that ia to say, the dasa of the earve. For axample,the envelopeof the Une
<Mt'-t86a'+8c<t+~=0,
where a, c, d, are linear functions of the co-ordmates,!a
plain1ya cnrve of the third dasa.
290. Now let us proceed in like manner with a surface.The equationof the tangent plane to a surface !a a functionof the three co-ordinates,whîch being connected by only onerelation (v!z. the equation of the sur&ce), thé équationof the
tangent plane, when most s!mpM6ec),containe two variable
parameters. The converse problem is to 6nd the envelopeofa plane whoseequation «'='0 contains two variable parameters
ft, ~8. The equation of any other plane answering to thevalues a+h, j8+~ will be
Now in the limit, when Aand k are taken indefinitelysmall,they may preserveany finite ratio to each other A=X~ Wesee thus that the intersection of any plane by a coBMcntiveone is not a deSmtoUne, but may be any line representedby
the equations se 0, d a + d u 0, where "de.thé eqnat!ona«!=<),y+~jg'='0,
where M indetenntnate.
But we aee atso that all planes consecutiveto u pass through
the pointgiven by thé equationsu=0, du =0, ~o=0.da d/3Fromthese three equationswe can eUminatethe parameters
a, ~S,and so findthe locusof aUthose pointa where a planeof
the ayatemis met by the aenea of consecutiveplanes. It ia
proved, aa in the last article, that the surface representedbythis eliminantis touched by u. If it be required to draw a
tangent plane to this surface through any point, we baveonlyto subatitutethe co-ordmatesof that point in the equationM'=0.The equationthen containingtwo indeterminatese and can
be aat!anedin an infinity of ways; or, as we know, througha given point an infinity of tangent planes can be drawn to
the surface,thèse planes enveloping a cone.
MMEOnVB MOMRTHSS OP CUBVB8. 8M
Suppose,however, that we elther consMer M coMtant,or as any definite functionof a, the equationof the tangentplaneMredneedto containa singleparameter,and the envelopeof thoseparticulartangent planeswhiehsatîs~ the assumedcon-ditionis a developable. Thus, again, we may see the analogybetweena developableand a curve. When a surface Mcon-sidoredas the locus of a number of pointsconnectedby a givenretatton, if we add another retation connectmgthe points weobtaina curve traced on the given surface. So if we considera surface as the envelope of a series of planes connectedbya single relation, if we add another relation connecting the
planes we obtain a developableenvelopmgthe given sarface.
291. Let us now seewhatpropertiecof developablesare tobe deducedfrom consideringthe developableas the envelopeof plane whoseequation containsa singlevariable parameter.In the firat place it appearsthat throughany assumed pointc<mbe drawn,not as before an infinityof planes of the aystem,forminga cône; but a definitenumber of planes. Thus if itbe required to find the envelopeof <Mt'+3&a'+3ca+J, where
<t,6,c, d represent planes, it is obviousthat only three planeaof the system can be drawn through a given point, sineeon
subatitatingthe co-ordinatesof any point we get a cubic for a.
Agam, any plane of the systemis cut by a consecutiveptane
in a definiteline; namely, the line Mc'O, j*'=0; and if we
eliminatea between these two equationswo obtain thé sur-face generated by all those lines, which is thé requireddevelopable.
It is proved, as at Art. 289, that the plane u touches the
developableat every point whichsattaSesthe equationsM==<~
-=0; or, inother words, touchesalong the whole of the line
of the system correspondingto M. It was proved (Art 107)that in général whena surfacecontainsa right Une the tangent
plane at each point of thé right I!neis din~rent. But in thecase of the developablethe tangent plane at every point is
the same. If a; be the planewhichtonches aU along the Mme
PROJECTIVE PROPERTtES 0F CURVE8.232
a:y, the equation of the surface can be thrown into the form
<c~+~=0 (aee p. 7S).*
292. Let us now consider three consecutive planes of the
system, and it is evident as before that their intersection satisfies
the equations M=0, y=0, yT=0. For any value of <t, theT,x da
point is thus determined where any line of the system is met
by the next consecutive. The locus of these points is got by
etinunatmg et between these equations. We thus obtain two
equations in x, y, e, one of them being the equation of the
developable. These two equations repreaent a eurve traced
on the developable. Thus it !s evident that starting with the
definition of a developable as the envelope of a moveable plane,we are led back to ita generation as thé locus of tangents to
a cnrve. For the consecutive Intersections of the planes form
a senes of lines, and the consecutive intersection of the lines
are a series of points forming a curve to which the lines are
tangents. We shall presently show that the curve is a cuspidal
edget on the developable.
It seemsunneeMsary to enter more Mty hto the subject of envelopesin general, since what M said in the text apptiM equally if «, instead of
representing a plane, denote any surface whose equation ineludes a variable<~t<
parameter. Monge calh the earve w = 0, 0, in which any surface of
the system M intersected by the eonMcutiTe, the <!&<n'<M!<o'M(<<!of the
envelope. For the nature of thls curve depends only on the manner inwhieh the ~tmabke x, y, t enter inta the ~mctioa u, and not on the manner
in whichthe constants depend on the parameter. Thus when « representsa plane, the oharactcrMtic ia alwayaa right line, and the envelope M the
Ioe<Mof a system of right lines. When u repretent* a tphere, the eha-
racteriatM,being the intersection of two coMecuti~e sphères, is a cirele
and the envelope h the !ocu< of a system of cirete~. And so envetopeein general may be divided into &mi!ies according to the nature of the
characteristic.
t Monge bas called this the "arête de MbroaMement," or "edge of
regresmon"of thé developable. There is a similar earfe on every envelope,
namely, the locus of points m which eaeh ehaMeteriatie'*h met by the
next conaecutive. The part of the eharaeteri&tie onone aide of thie earve
generates one sheet of the envetope, and that on the other aide generateeanother aheet. The two eheets touch along this curve which M their
MMMECTtYt! FMPMTÏË8 0F CUBYE8. 233
293. Four consecutiveplanes of thé system will not meet
in a point udess the four conditionsbe nualled «==0, y=0,
,~=0, -~=*0.It is in general possible to find certain
values of et,for whieh this conditionwill be aatisSed. For ifwe eliminate<e,y, e, we get the conditionthat the fourplanes,whose equations have been juat written, ehati meet in a point.This condttion is a 6mction of a; and by eqn&tmgthis iunctionto nothing, we shall in general get a determinate number ofvalues of a for which the condition is satisfied. There aretherefore in general a certain namber of points of the systemthrongh which four planes of the system pasa; or, in other
words, a certain number of points in which three consecutivelinesof the systemintersect. We shallcall thèse, as at .S~~r2%t<MCMfMa,p. 28, the <<a<!MM~points of the system; sincein this case the point determined as the intersection of twoconsecutivelines, coincideswith that determined as the inter-sectionof the next consecutivepair.
ReciprocaUy,there will be in general a certain number of
planes of the system which may be called aMtCMryplanes.These are the planes which contain four consecutivepointsof the system for in sucha casethé planes 183,234 evidentlycoincide.
294. We shallnow showhow,fromPl&cker'sequationscon-
necting thé ordinary singularitiesof planecurves,* Mr. Cayleyf
commonlimit andMa etNpMale~e of theenvelope.Thusin thecaseofa conethe p<rt!)of the generatinglinesonoppositesideaof thevertexgenerateoppositetheetsof the eonefandthe cuspidaledgein thiecaseredaceaitseiftoa singlepoint,namely,thevertex.
TheMequationsare as followe:BéeJS%'A<r.MtMCttrMt,p. 91.Let be the degreeof a curve,tta ctaM,8 the numberof its doublepointa,ttMt of!t8doubletangents, thé numberof it$cMp$,that ofitapointsof Memen 1then
<\ AJt 0-. t. t\ 0~- Q.
MMEOnvNHtOMKnBaMf CORVEE884
ÏMMdedaced equations eonnec~mgthe ordinarysiNgatanuesof
devetopablea. We sh&Ufirat make an enumeration of these
singalanties. We speak of the "points of the system,"the"lines of the System," and thé "planes of the system" Mex-
plained(Art. 119).Let m be the number of points of the systemwhich lie in
any plane; or, in other worde, the degreeof thé curve whieh
generatesthe developable.Let Mbe the namber of planes of the ~stem whieh c<mbe
drawnthrough an arbitrary point. We have proved (Art. 291)that the number of such planea is dennite. We sh&Ucall thisnumber the e&tMof the aystem.
Let r be the number of lines of the ayatemwhich iateiseotan arbitrary right Une. It iB plain that if we form the con-
di hadu
d -1 b 1..dition that M,Tand any assumed right line may inteNect,
the remtit will be a funetion of whieh being equated to
nothing gives a deSnIte nnmber of values of a. Let f be thenumber of solutions of this equation. We ahaU call thianumber thé f~H~ of thé syatem, and we shaUshow that aU
other MngatMttIesof thé system can be expreased in termsof thé three just enumemted.
Let a be the number of stationary planes,and ~3the nnmberof stationary points (Art. 298).
Two non-cooseentivelines of the Systemmay intersect.When this happons we caU the point of meeting a "pointon two lines," and their plane a "plane through two lines."
Let at be thé number of "pointa on two lines" which lie
in a given plane, and y the number of planesthmugh twolines" vhich pass through a given point.
In likemannerwe shallcaUthe linejoining any two point*of
thesystema Unethroughtwopoints,"and thé intersectionofanytwo planesa I!nein two planes." Let gbethe namberof lines
in twoplanes" which lie in a given plane, and Athe numberoflinesthroughtwo points" whichpass throngh a given pointThe developablebas other singulantles whichwill be dater-
mined in a subsequent chapter, but these are the singularitieswhichPHtoker'eéquations (note,p. 288) enableus to determine.
HMMMTIYB MOPBttTtES or CUNVES. 28S
296. Consider now thé sectionof thé developableby anyplane. It is obviousthat thé pointeof thia carve are thetraces
on its plane of the lines of the System,"white the tangentlines of the section are the traces on its plane of the "ptaneaof the system." The degree of the section M therefore f,since it is equal to the numberof pointe in which an arbitraryline drawn in its plane meetsthe section, and we have sucha point whenever the Unemeetsa *me of thé system."
The chas of the sectionia plainly M. For the number of
tangent Imes to the sectiondrawnthrough an arbitrary pointis evidentlythe sameas the numberof "planea of the system"drawn through the samepoint.
A dcaMe point on the section will arise whenever two« lines of the System" meet the ptane of section in the same
point. The number of such points by definition is <& The
tangent lines at such a doublepoint arc mtta!lydistinctbecausethe two planes of the system coïteapondingto the lines of the
system interaecting in any of the points a; are commonlydinerent.
The number of doubletangents to the section !s in like
manner s!ncea double tangent anses whenever two planesof the systemmeet the planeof sectionin the sameline.
The mpointsof thé systemwhiehlie in the planeof sectionare cnsps of the section. For they are doublepointa as beingthe interaection of two lines of the system; and the tangent
planes at these points coincide,smcothe two consecutiveUnes~Intersecting in one of the points m, lie in the same plane ofthe system. This proves, what we have a!ready stated, thatthe carve whosetangents generate the developable!s a cospidaledge on the developable;for it is such that every plane meetsthat surface in a section which bas M casps the points wherethe same plane meetsthe curve.
Lastly, we get a point of inflexion(or a stationary tangent)wherever two consecutiveplanesof the systemcoincide. Thénumberof pointsof inflexionis therefore&
We are to subsutate then in the fbrmoia, note p. 233,
MOJBCTIVB MOHSSTÏES 0F CURVES.286
296. Another system of equations is found by consideringthe cone whose vertex is any point and which stands on the
given curve. It appeara at once by conaideringthe sectionof a cône by any plane that the same equations connectthedoublepoints,double tangent planes, &c. of cones, whichcon-nect the doublepoints,doubletangents, &c.of plane curves.
The edges of thé cone which we are now consideringare
the Unesjoining the vertex to all the points of t!te System;and the tangent planesto thé cone are the planes connectingthe vertexwith thé linesof the system, for evidentlythe planecontainingtwo consecutiveedgesof the cône must containthe
line joiningtwo consecutivepointsof the system.The degree of the cone is plainly the same as the degreeof
the curveand is thereforem.The class of the cone is the same as the number of tangent
planesto the cone which pass through an arbitrary line drawn
through the vertex. Now since each tangent plane containsa line of the system, it followsthat we have as many tangent
planes passing through the arbitrary Une as there are lines
of the system which meet that line. The number sought isthereforef.*
A double edge of the cône anses when the same edge of
the cone passes through two points of the system, or S=A.
The tangent planes along that edge are the planes joiningthe vertex to the Unes of the system which correspondto
each of these points.
It Measy to see that the c!&Mof thh cône is the mme as the dentéeûfthe developablewhieh ta the reciprocal of the points of the given tyttem.HeXee,Me <&~Mof <jte<<tce&!p<!M~~MfwMby the tangents goany <'«tw
il <i5<M<Met <&<<&~e< the <&«<t!poMt?&«!&MMe fte~oM~ me
~ot«<<of <<«<ettfw, see note, p. 124.
PBOJECTtVN PROPERTtN) 0F CUBVE8. 287
A doublo tangent plane will arise when the same planethrongh the vertex containstwo lines of thé system; or T=y.
A stationary or cuspidaledge of thé cone will only existwhen there ia a stationary point in the System; or <c'=~8.
Lastty) a stationary tangent plane will exist when a planecontainingtwo consecutivelines of thé system passes throughthe vertex; or i==M.
Thus we hâve /t=M, y=~ S=A, T'=~, <=' '=?.Hence by thé fbrmuhe(note p. 233)
PI&cker'aequations enableus, when three of thé singularitiesof a plane curve are given, to determineall the rest. Nowthree quantities r, m, n are commonto the equations of thisand of the last article. Hence, whenany three of <~ s&~M-&)T<MMwAM&«? have eM«)M'Mt<e<~of a curve in t~Mce,are
given,all ~< t*e<<can &e~oM)!<
297. To illuatrate this theory, let na take the developablewhich is the envelope of the plane
where t is a TMMtMeparameter, a, b, c, &c. represent planes,and k M any integer.
The dasa of this systemMobviously&,and the equationof the developable being the discriminantof the preceding
equation, !t8 degree is 2(&–1); hence<'==2(~–1).A]so it is eafy to see that this developablecan have no
atat!oNaryplanes. For in gcneral if wo compare coeOMentsin the equationsof twoplanes,three conditionsmust be satisSedm ordor that the two planes may be identical. If then we
HMMN'nVB PN(HRMMB9 OF CUKVE8.238
attempt to detenNme <eo that any plue may be ident!ca!with the consecutiveone, we Cndthat we have threoconditionsto satiafy,aud only one cotMttmt<tLtour disposât.
H~vingthen K=&, f ='2 (t 1), a = 0) the equ<tt!onsof the!aat two articles enable us to determine the rmaining NDga-M~es. The result is
The greaterpart of these values can be obtained independentlyas at .B%~ ~%M<C')(nxM,p. 94. But in order to economize
spacewedonot enter into dettula.
298. The case consideredin the lut article, which is thatwhen the variable parameter entera only ratIoNaByinto thé
equation, enables us to verify easily many properties of de-
velopables. Since the system «=0, ~=0îs obvioualyre-
ducible to
it followsthat ocis itae!f a plane of the ayatem (namely,that
coneepondingto thé vatue <='oo), aiMBthe correspondingline,and atc the corresponding point. Now we know from the
theory of ~acrmunamta(BéejB%)' ~~<~n)t, p. 47) that the
equationof thé developableis of the &mt +~"0, whereis the dMcnminantof u when in it a is made=0. Thus we
verify what was stated (Art. 291) that a touches the deve1op-able along the whole length of the line ab. Further, isitse!f of the form %+<?~ If now we consider the aeetMncf the developableby one of the planes of the ayatem (or, in
other worde,if we make <t'=0 in the equation of thé develop.aHe), the MctmneomMtaof the !lM ab twice and of a curve
289MMMECTtVEPROPNtTïBS0FCUBVM.
ef the degree f 8 aad this c<)rve(asthe form of the equation
shows)touches the Uneab at the point <<~c,and conteqnentlymeeta it in f–4 other points. Thèseare &Upoints on two
lines," being Ae points where the line ab meetaother linesof the system. And it !a generallytrae that !f r be thé rsnkof a developable«M&line of ~e ~<Mt ttM~ r–4 otherliMs
of ~e ay<<MK.The locus of the<epointsformea doublecarre
on the developable, the degreeof whichM .c, and the other
properties of which will be given in a subsequent chapter,where we shall aho determinecertain other singularities ofthe developable.
We add here a table of the singularitiesof some ape<aalMctMMof the developable. The reader, who may care toexaminethe eubject,will find no great di<Ecnt<yia eatabliahingthem. 1 bave given the proof of the greater part of them,
Cambridgeand Dublin JM<~<MOt<M<~JoMMM?,VoLV.,p. 24.
Sectionby a planeof the system
CLA88!FïCATKHt0F CMVE8.240
SECTIONH. ca~ASSIFICATMHf0F CURVES.
299. The followingenumeration rests on the principlethata curve of the degree r meets a surface of the degreep in
~r points. This is evident when the curve ia the complèteintersection of two surfaces whose degrees are M and n.For then we have )'=MH and the three surfacesintersect in
mnp points. It i< true ako by definitionwhen the surfacebreaks up into p planes. We shall assume that, in virtueof the law of conttNuity,the principle is generally true.
The uae we make of the principleis this. Suppose thatwe take on a curve of thé degree f, as many points as aresufficient to determinea surface of the degreep; then if thenumber of points so assumed be greater than jM', the surfacedescribedthrough thé points must altogether containthe cnrve;for otherwise the principlewouldbe violated.
We assume in thia that the curve la a proper cnrve of thé
degree y, for if we took two curves of the degrees m and n
(where Mt+M=~-), the two together might be regarded as a
complex cnrve of the degree f, and if e!~ef lay altogetheron
any surface of the degree of coursewe could take on thatcurveany number of points commonto the curve and surface.AUthis will be sufficientlyillustrated by the exampleswhichfollow.
300. ï~&ao~MM.For throngh any two points of a line of the firat degreoand
any assomed point we caa describe a p!an6 whîch must alto-
gether contain the line, since otherwisewe ehouldhavea !moof the first degreemeeting the plane in more points than one.In Hke manner we can draw a second plane containing the
line, which must therefore be the intersection of two planes;that ia to say, a right line.
TX~e M MOproper ?MMof the MCOMf?degree tM<a conie.
Throagh any three points of the line we can draw a ptane,which thé preceding reasoning showsmuât altogether containthé line. The Ime muet therefore be a plane curve of theseconddegree.
Ct.A68t!'ïCATM!{ 0F CURTE8. 241
B
The exception noted at the end of the !axt article would
occar if the line of tho second degroe eonsisted of two rightlines not in the same plane; for then the plane throngh three
points of the system would only contain o?Mof thé right lines.
In what follows we shall not think it necessary to notice this
again, but shall epeak only of proper curves of the!r respectiveorders.
301. c«M)e of the third <e~ must either &ea planeCMSMof the JM!(M? M~MechoMof ttpo ~tMM~M~as cxplained,Art. 885.*
For through aeven pomttt of the carva and any two othcr
points describe a quadric; and as before, it mast altogethercontain the eurve. If the quadric break up into two planes,the curve may be a plane curve lying in one of tic planes.As we may evidently have plane curvea of any degree weshall not think it necessary to notice thèse in subsequent caaea.
If then the quadric do not break up into planea, we can draw
a second quadric throngh the seven points, aad thé intersection
of the two qnadncs includes the given cubic. The completeintersection being of the fourth degree, it must bo the cubic
together with a right line; it is proved therefore that the
only non-plane cubic is that explained, Art. 285.
802. The cône contammg a curvo of the <n"' degreo andwhoae vertex ie a point on the carve, is of the degree <M 1hence thé cône containing a cubic and whose vertex is on the
curve is of the second degree.f We can <AtMdescribe a f:eM<e<~
Non-planecurveaof the thM degreeappear tohavebeenButnoticed
by M3MM in hie .Boty!eK<tt<!<4<<e«<tM,1827. Some of their most
importantpropertiesare given by M. Chasles ia Note xxxtn. to Hs
~Mt~« JBM<M~t«,1837, and in a paper in LiouvINe'tJ<)Mf)M7for
t8<7, p. 397. More recently the propertieeof thèse carfes have beentreated of by M. SchjSter, CM!f,Vol. t,ï][.,and ny ProfeMorCremonaof tîihn, C~<<!it,Vot.l.vm., p. t38. ConaideMMeuee has been madeofthe latter paper in thé articleswhiehimmediatelyfollow.
t M. Chasleshastilysaid that converselythe locusof the vertex ofa côneofthe seconddegreepaaaittgthroughsix pointa,is thecubicthroughthese points. But as Mr. Weddlepointed out, <X<M~M%wa)t<<JMMt
CLASSIFICATION0F CURVES.M2
«tMe ttroM~A j~teeMpoints. For we can descrUioa côneof the second degree of whieh the vertex and nve edges are
given, s!nce evidently we are thus given five points in the
sectionof the corneby any plane, and eu thus determinethatsection. If thon we are given mxpoints a, &,c, <?,e, wo
can describe a cone having the point a for vertex, and thelines a~, ce, ad, <M,af for edges; and in like manner a cone
having b for vertex and thé lines &t, bc, M, for edges.The intersectionof these coneaconstatsof thé commonedge aband of a cubic whieh is the required carve passing thronghthe six points.
The theorem that the Unes joining six points of a eoMoto any seventh are edges of a quadriccône, teads at once tothe followingby Pascal's theorem Thclines of intersectionof the planes 718, 745; 783, 756; 734, 761 lie in one plane."Or in other words, thé points where the planes of three con-secutive angles 567, 671, 712 meet the oppositeaides lie inone plane passing throngh the vertex 7. Convereelyif thishe true for two vertices of a heptagon it is tme for all the
rest for then these two vertices are verticea of cônes of the
second degree containing the other points, whichmnst there-forelie on the cnblcwhichis the intersectionof the cones.
303. JL cubictracedona ~pe~toM~ of one<Xee<<K<~all its
~ene~N~M'aof <Me<ty<<eM<MMe,and <5<Meof other~<.tHKtwice.
Any generator of a quadric meets in two points ita cnrveof intersectionwith any other quadric,namely, in the two pointswhere the generator meets thé other quadric. Now when the
~&<t<t)tM<M<!<JOMHM~VoL v., p. 69, the locusof the vertex is not acurvebut a aur&ce,namely,that obtainedby etimtMtin~ p, betweenthe four diaëraittah of <S+\0'+~P'+f~ where~S,P; ]f; W are anyNtt&ceathmt~httMMxpomtt.
The toeue of the vertex of a côneof the second«tdar whiohpMM<through seven pointail a earw and Mcf the Mxthorder. When eightpointaare givenfourconeecan be desenbedthroughthem. Seeappendix"on the order ofayatemaof equations."
M. Cremonaaddethat whenthe aixpointeaM&tedand the seventhvaaaNe,thMplanepMMtthrougha &tedchordofthe cubio.
CI.ASS!nCMM!t Or CMVM. 24&
R~<)
intersection consista of a right line and a cubic, it is evident
that the generators of thé Rameaystem aa thé line, since theydo not meet the line, must meet the cnMo in the two pointa;white the generatom of the oppoaite system, sinco they meet
the line in one point, only meet thé cubic in onc other point.Conversely we can describe a system of hypcrboloids throngh
a cubic and any chord wh!ch meets it twice. For take
seven points on the curvc, and an eighth on the chord joining
any two of them; then through thèse eight points an InSnityof quadries can he described. But sinee threo of the» pointsare on a right line, that Une most be common to all the
quadnea~aa must also the cubic on which thé aevcn pointa lie.
304. The question to nnd tho envclope of 3&<*+ Sct d
(where a, &,o, d represent planes and t ia a variable parameter)is a particular case of that discussed, Art. 297. We have
Thus the System is of the same nature as the rcciprocal ~atem,tmdall theorema respecting it are conseqnently two-fold. The
system being of the third degree most be of thé kind we are
considering; and this also appeara from the equation of thé
envelope
for it is easy to see that any pair of the sur&cea a~-Sc, b'- ac,
c'–M, hâve a right Une common, wh3o there is & cubic
common to all three, which !s a double lino on thé envelopo.It appoars from tho table jnst given that cvery plane con-
tains one line in two planes"; or that thé section of the
developable hy any plane bas one double tangent; white re-
ciprocally throngh any point can he drawn one Une to meet
thé <mMotwice; the cone thcre&re, whose vertex is that point,and whieh stands on the carve bas one double point; or in
other worda, the c!<McM jM~ec<e~on any plane ttt~ o OMM:
%<!C:*H~'0 double point.The three pointe of inflexion of a plane cubic are in one
right line. Now it wu proved (Art. 296) that thé points of in-
flexion correspond to the three planes of the system whioh can
be drawn through the vertex of the cone. Hence the three
CLA6StFICATM!t 0F CURVES.244
pointsof the systemwhich correspondto the three planeswhichcan be drawn through any point 0, lie in one plane passingthrough that pointa
Further it ia known that whena plane cubic has a conjugatepoint,its three pointsof inflexionare real; but thatwhenthe cubicbas a doublepoint, the tangents at which are real, then two of
thé points of inflexionare imaginary. Henceif the chordwhichcan be drawn through any point 0 meet the cubic in two real
points, then two ofthé planesof the systemwhichcan bo drawn
through 0 are imaginary. ReciprocaUy,if through any linetwo real planes of the system oan be drawn, then any planethroughthat line meets the curve in two imaginarypoints,and
onlyone real one.t
805. These theorema can also be easily established alge-
braical!y for thé point of contactof the plane< 8&<*+ 3c( d,
being given by tbeequations <!(=&,bt=o, c(=:<~may be denoted
by the co-ordinates a=l, &==f,cc=~, <=!< Now thé threevaluesof t answeringto planes passing through any point are
givenby the cubic<t'~ 3&'<'+ 8c'< <f= 0, whence it is evidentfromthé values just found, that the points of contact lie in tho
planedd 8&'c+ 3c'& tfa =0. Bat this plane passesthroughthegiven point. Hence <~M~efMe<«MtO/M6~&MMao/e<ya<e)H~«Min plane 0~the <!<M~Mp<MK)~points. The equation JMtwritten is unaltered if we Interchangeaccented and nnaccentedletters. Hence if a ~OMt< <Mtkeplane co~espM~M!~toa
point J9, J? will bein theplane c<M'yMpoM<~M~? And again,the planes which correspond to all the pointsof a line .<iBpassthrough a 6xed right line, namelythe intersectionof the planescorrespondingto and J9. The relation betweenthé linos is
plainly reciprocaL To any planeof the systemwill correspondin this sensethe correspondingpoint of the system andto a linein two planes correspondsa chordjoining twopoints.
The three points where any plane At + Bb+ Cb+ JMmeets thé curve have their <'s given by the equation
ChMiM,Z~MM~,1867. SehrOtcr,<< VoLLVï.t JoacUmethal,0'VoI.î.T[.,p.M.CKmona,(~<<h,Vo!.i.vm.,p.MC.
CLA6fHHCATMN0F CURVEt). 24B
JW+ Ci!'+JB!f+~==0, and when this is a perfect cube, the
planeis a planeof the system. From this it followsat once,asJoachimsthal bas remarked, that any plane drawnthrough theintersection of two real planes of the system meets the curvein but one real point. For in sncha casethc cubicjust writtenis thé sumof twocubesamdhMbut one'realfactor.
306. We hâve seen (Art. 124)that a twisted cubic is thelocus of the pôles of a nxed plane with regard to a systemof quadncs having a commoncnrve. More generally aucha curve is expresed by the resnit of the elimination of X
betweenthe systemof equationsXa's a', X~=& X<; c'. Nowsince the anharmonicratio of four planeswhose equationsareof the form \oc=<x', X'<t=a', &c. depends only on thecoefficients &c. (sce C~hM, Art. 66), this mode of
obtaining thé equation of the cubic may be interpreted asfollows: Let therebe a systemof planes through any line aa,a homographiesystemthrough any other Unebb, and a third
through ce', then the locus of the intersection of three corre-
sponding planes of the systems!s a twiated câble. The lines
aa', ?', ce' are ovidentlyUnesthrough two points, or chordsof thé cubic. RecIprocaIIy,if three right Unes bo homo-
grapMcaUydivided, the plane of three correspondingpointsenvelopes thé developablegeneratod by a twisted cubic, andthé three right Unesare UUnesin twoplanes" ofthé system.
The tine joining two correspondingpoints of two homo-
graphtcally dividedlines, touches a cornewhen thé Unes arein onopiano, and générâtes a hyperboloidwhen they are not.Hence givcn a series of points on a right line and a homo-
graphie aorieseither of tangents to a conic or of genoratorsof a hyperboloid,the planes joining each point to the corre-
spondingUne envelopea developableas above stated.
Ex. If the four&eesofa tettahedionpan throughBxedlinea,andthree verticesmovem &)tedBne<,thelocusof the temaimngvertextaa twistedeuMc.Any numberof positioneof the bMeforma ayatemof planeswhiohdividehomo~rapMcaUythe threeJinMon whiehthecomeNof the baMmove,whenceit followathatthe threeplaneswhiehînteMeotin the vertexare correspondingplacéeof threehomographiesysteaM.
246 CLAS6!FÏCAT!ON 0F CUBVES.
807. From thé theorems of the last article it followscon-
vemelythat thé planesjoining four nxed points of the systemto any variable line throngh two points form a constant an-
harmoniesystem" and "four nxed planes of the systemdivide
any t!mein two planes' in a constant anhamonie ratio." Itis very easy to prove thèse theorems independently. Thusweknow that the sectionof the developableby any plane ofthe system, cons!stsof the correspondirigline a of the systemtwice, together with a conio to which aIl other planes of the
system are tangents. Thus then the anharmonic property ofthe tangents to a conic shows at once that four planes eut
any two lines in two planes, AB, AC in the same anharmonic
ratio; and in like manner J[<7ts eut in the sameratio as CD.As a particular case of these theorems, since the Unesof
the system are both lines in two planes and Unes throughtwo points ;~Mr~<!e<f~at!Mof tAe<ty<<eMeut all <~ hKMof<&e~<oK & the «MMoH&tnK~KMratio; and <~ ~«MMjoMM~~M)'~ee<~~<Wt&of the <~<<SMMto <tHthe ?MM~of tke ~t(MMarea CMM&Mt<<MAanM<M!M~~feM.
Many particular inforences may be drawn from these
iheoremsas at CiMMt,p. 278, which see.Thus consider four points a, ~8,'y, 8; and let ua express
that the planes joining them to the Unes a, b, and 0)8,eutthe line 'yohomographicaUy. Let thé planes A, B meet 'yo in
points <, (. Let thé planes joining the line a to /8, and theMne&to a meet y8 in &,< Then wehâve
If the pointa t, &' coineide, it followsfrom thé nmt equationthat the points &, (' coincide, and from the second tbat thé
pointst, < y, o are a harmonio system. Thus we obtainProf. Cremona'a theorcm, that if a series of chords meet théline of intersection of any plane d with the Hne joining the
correspondingpoint a to any line b of the aystem, then they
It ie oftenconvenienttodenotethe phuteoof thesystembycapitaltetteK,the correspondinglinesby italica,and the correspondingpointsbyOrethletters.
CL&66tFM~HON 0F CCMES. 247
will atso meet the Une of intersectionof the plane B withthe line joinicg to a; and will be eut harmonicallywhere
they meet these twoUnesand wherethey meet the cnrve.The reader will have no digicnlty in aeeing when it will
happen that one of thèse Unespassesto infinity,in which casethe other line becomesa diameter.
308. We have aeen that the eectiom of the developableby the planes of the systemare conics. We may thorefore
inTes<jga.tethe loom of thé centres of these comcS)or more
generally the locas of the poleawith respect to these conicsof thé interaectionaof their planes with a Sxed plane. Sincein every plane we can draw a ~I!ne in two planes" we maysuppose that thé nxed p!ane paMesthrough the intersectionof two ptanes of the system ~4,B.
Now conalderthe sectionby any otherplane 0, the traceson that plane of and B are tangents to that section, andthe pole of any Ime tbrough their interseetionlies on their
chord of contact, that M to say, lies on the line joining the
points where the unes of the systema, meet C. But s!nceall planes of thé Systemcut the lines a, & homographtcaUy,the joininglines generatea hyperboloidof one sheet,of which
a and b are generators. However then the plane be drawn
through thé Une ~B, the loeas of polo is this hyperboloid.But further, it is evident that the pole of any plane throughthe intersectionof B lies in the planewhich ia thé harmonic
conjugateof that plane with respect to those tangent ptanes.The locusthereforewhichweseek is a plane conic. It is plainalsofromthe constructionthat since the pôleswhen any planeJL+\B is taken for the nxed plane, lie on a conio m thé
plane A XB; converselythe locus when the latter is takenfor Ëxedplane is a conioin the formerplane*
809. In conclusion,it ia obviousenough that cnbics maybe dividedinto four speciesaccordingto the dtmerontsectionsof the oorve by the plane at infinity. Thus that plane may
CLASStFtCATtOX0F CURVE8.248
either mcet the curve m three real pointa; in one rcat andtwo imaginary points; in «ne real and two coincidentpoints,that is to say, a line of the system may ho at infinity or
lastly, m three coincident points, that M to say, a plane ofthe system may be altogether at infinity. These specieshâvebeen called the cubicalhyperbola,cubicalellipse,cnbicalhypersbolic parabola, and cubical parabola. It is plain that whenthe curve bas realpoints at infinity, it bas branchesproceedingto infinity, the !ines ofthe system correspondingto the pointsat infinity being asymptotes to tho curve. But when théline of the system!s itsetf at infinity as in the third and fourth
cases, the branchesof the curve are of a parabolic form pro-ceeding to infinitywithout tending,4o approach to any Bnite
asymptote. Since the quadric cones which contain the curvebecome cyl!ndemwhen their vertex passes to infinity, ît !s
plain that three quadric cylmders can be describedcontainingthe eurve, the edges of tho cytmders being parallel to thé
asymptotes. Of conrse in the case of the cubicalellipse twoof these cylinderaare imaginary: in the case of the hyper-boMcparabola there are only two cylmders, one of which is
parabolic, and in the case of the cubical parabola there isbut one cylinder which ia paraboitc.
It follows from Art. 804 that in the case of the cubical
ellipse the plane at infinity contains a real line m two planes,whichis imaginaryin tho case of the cubicalhyperbola. Thatis to say, in the former case, but not in the latter, two planesof thé system can be paraReL From the anharmoniopropertywe infer that in the case of the cnbical parabolathree planesof the system dividein a constant ratio all thé lines of the
system. In this case all the planes of the system out the
developable in parabolas. The system may be regarded asthé envelope of z<8~<*+8~<–<~ where d is constant. Forfarther détailswe refer to Prof Cremona'sMemoir.
310. We proceednow to thé dassincationof curvesof higherorders. We have proved (Art. 299) that through any ourvecan be describedtwo surfaces, thé lowest values of whose de-
grees in eaeh case there is no dimculty in determining. It
CLA8MF!CATKM! OP CURVES. 249
is evident then on the other hand that if commencing withthe aimplestvalues of and v we discuss all the dtSerentcases of the intersection of two surfacesvhoae degreea are
and v, we shall inctudeall poMiMecurves up to the r"* order,the value of this limit f being in each caseeasy to find when
/t and f are given. With a view to auch a d!acnMtoawe
commenceby inveatigatingthe charactensticsof the curve ofintersection of two NM&ceB.*We have obviously Mt==~f,and if the surfacesdo not touch, as we shall suppose theydonot, their curveof intersectionbasno multiple pointa (p. 95),and therefbre ~8=0. In order to determine completely thecharacter of thé system, it is necessary to know one moreof its singatarities, and we ehooseto seek for r, the degreeof the developablegeneratedby thé tangents. Now th!a de-
velopableis gotby eliminatingz'y'f!'betweenthé four equations
0~0, P"=0, Za:+J~+~+J~=0, Z'ie~ ~'y+~+P'M~O,
whereZ, JM,&c. are the Crst differentialcoe~oients. Thèse
equationsare respectivelyof the degrees /t, f, /t-l, f–1:
and since only thé last two containa~ theae variables enterintothe resalt in the degree
which !s evidently of the degree ~+y-2. This denotea a
surface which ia thé locM of the points, the intersection of
whose polar planes with respect to U and F meet the arbitrarylino. And the points where this locus meets the curve FF
The theory explained !mthe remainderof thie Mettott Mtaken from
a paper dated July, 1049, which 1 pabMahedin the ChMM~< <tM<<JPttM'M
JM«~M<t<tCtt<Jo<M''M~,VoLv., p. 23.
260 CLA8S1NC&TÏONOF CUBVE8.
311. We verify this result by determining independentlyA thé number of "Unes through two pointa" which can pamthrough a givcn point, that !a to say, the number of lineswhich can be drawn through a given point so as to passthrough two pointsof the intersection of U and P. For this
purpose !t ia necessary to remind the reader of the method
employed at the foot of p. 86 in order to find the equationof the cone whosevertex is any point and whichpassesthroughthe intersectionof U and Y. Let na supposethat thé vertexof the cone is taken on the curve so as to have both U andF=0 for the co-ordinates of the vertex. Then it appearsfrom p. 86 that the equation of the cone ia the result of elimi.
nating Xbetween
Thèse equationsin are of the degfees~-l, y 1 8 U,8*~&c. containthé co-ordinatesa; a~e in the degreea 1,1~-2,2,<&c. A specimen terra of the result M (SO')'Thus it appears that the result contamethe variableso:~ inthe degree !l+)'(~t-l)==/tf-l; while it oontama a:y<in the degree (/<-1) (f– 1). Every edge of this cone of the
degree ~f–l, whose vertex ia a point on the curve, is ofcourae a "ne through two points." If now in thia cônewe consider the co-ût<dmateaof any point a~e on the coneas known and a;y.e' as sought, this equation of thé degree(~ -1) (y -1) combinadwith the equations Uand V determine
CLMMFtCATMN0F CURVES. 251
the "pointa" belongingto all the "!ines through two pointa"which campass thronghthe assumedpoint. The total number
of sach points M therefore ~t)'(/t–l)(f-l), and the numberof lines through two pointa is of course half thia.
The number determinedin thia article, 1 ca thé numberof apparent double points in the intersectionof two snr&ces,for to an eye placed at any point two branches of a carve
appear to interaect if any une drawn throngh the eye meetbothbranches.
812. Let us now considerthe case when the curve FFhaa also actual doublepoints; that la to say, when thé two
surfacestonch in one or more points. Now in thie case, thennmber of a~p<Mw:<doublepoints remains preciselythe sameas in thé lut article, and the cone, standing on the cnrveof intersectionand whosevertex is any point, has as double
edges the lines joining tho vertex to the points of contact inaddition to the number determined in the last article. It
is easy to see that the investigationof the last article does
not include the lines joining an arbitrary point to the pointsof contact. That investigationdetermines the namber of caseswhen thé radius vector from any point bas two values thesame for both surfaces, but thé radius vector to a point ofcontact bas only one value the same for both, since the pointof contact is not a double point on either sarface. Everypoint of contact then adds one to thé number of double edgeaon the cone, and theretoro diminishesthe degree of the de-
velopableby two. This might a!so be dedncedfrom Art. 810since the surface generated by the tangents to the curve ofintersection must Includeas a factor the tangent plane at a
pointof contact, sinceevery tangent line in that plane touches
the cnrveof intersection.If the surfaceshave stationarycontactat any point(Art. 129)
the une joining this point to the vertex of the cone isa cuspidaledge of that cone. If then the surfaces tonch m t points of
ordinarycontactand in ~8ofstationarycontact,we have
CLASSIFICATIONOf CCRV~S.232
and the reader can calculate without diScdty how the other
numbeMin Art. 8t0 are to be modiSed.We can hence obtain a limit to thé namber of points at
whiehtwo surfacescan touch if their intersectiondo not break
up into curveaof lowerorder; for we have onlyto sabtract thénumber of apparentdoublepoints fromthe maximumnumberofdoublepoints whicha cnrve of thé degree /tf canhave (J~~rPlane Ctf)W<,p.St)~
813. We shall now show that when the cnrve of Inter-section of two surfaces breaks up into two simpler curves,the charactemt!cs of these curves are so connectedthat whenthose of thé one are known those of thé othcr can be found.It was proved (Art. S11) that the points belonging to thé
!mesthrough two points" which pass through a givcn pointare thé intemecdon of thé curve 01~ with a surface whose
degree is (p -1) (~ 1). Supposenow that the curve of inter-sectionbreaksup into two whose degrees are m and M',where
M+M'==~tf,then evidently the "~vo points" on any of theselines must either lie both on the curve m, both on the curve
M', or one on one curve and the other on thé other. Let thenumber of Unesthrough two points of thé nrst curve be ~5,those for the secondcurve h', and let H be the nnmber of lineswhich pass througha point on each curve, or, in other words,the number ofapparent intersectionsof the curves. Consideringthen the points where each of the cnrves meet the surface
of the degree (~ 1)(~ 1),we haveobvioustythe equations
Thus whenMand A are known m' and A'eau be &)nnd. Totake an example which we have atready dtacnssed,let theintersection of two qnadrics conast in part of a right line
(forwhich M'=l, A'==0),then the renMmumgmtemectiommustbe of the thui degree m=3, and the equationabove writtendetermines A=Ï.
314. In like manner it wu proved (Art. 810) that thelocus of points, the intersection of whose polar planes with
C~ASNFICATMN0F CMVBS. 253
regard to F aad F meets an arbitrary line, !s a mrfaco ofthé degree /t+f-8. The 6tst carve mects this surface inthe t pointe where the curveam and m' interaect (since Uand V touch at these points) and in the r points for whichthe tangent to thé curvemeetsthe arbitrary line. Thus then
an equation which can easily be proved to followfrom thatin the tast article.
The intersectionof the coneswhich stand on the carvca
M, M' consisteof the t Unesto the pointa of actual meetingof the curves and of the J? Unesof apparent intersection; andthe equation .B+(==mm'is easily vcrifiedby nsim~the values
just found for F and f, rememberingalso that M'=~f–M,<-=)?(<?- 1)-8&.
315. Having now estaMishedthe principleswhich we shaH
have occasionto employ,we resume our enumeration of thodifferent speciesof carves of the fourth order. Every quarticcurve ?Maon « y«M?ne. For thé quadric determined by nine
pointaon the carvemust altogethercontainthe curve(Art. 299).It is not generallytrae that a secondquadric can be described
through the cnrve there are therefore two principal familiesof quartics,viz. thosewhichare the intersectionof twoquadrics,and those through which only one quadric can paas.* Wecommencewith the cttrveaof the first family. The character-istics of the intersectionof two quadricswhich do not touch
are (Art. 810)
Several of thèse resnits cambe establlshed independently.Thne wehave given (Art. 160)the eqtmt!ouof the developablegenerated by the tangents to the curvowhîch is of the eighth
degree. It ia there provedalso that the developablehas in
each of the four principal planeaa double line of the fourth
The Mhtenceof thissecond&mityof quarticswas,1 believe,~Mt
pointedout in the Memoirahwdyreferredto.
254 CMSSmCA'notfOPCCBVE8.
order, whence a?= 16.* Again, it is shown, p. 123, that thé
equationof the oscalating plane is ~"F=j8'P, which containsthe co-ordinates of the point of contact in the third degree.If then it bc required to draw an osculating plane throngh
any assumed point, the points of contact are determinedas
the intersectionsof the curve <7F with a surface of the third
degree, and thé problem tberefbro admits of twelve solutions;in =12. Lastly, every generator of a quadric containing thecnrve is ovidently a "une through two points" (Art. 808).Since then we can describe through any assumed point a
qaadricof the form XP, the two generatorsof that quadricwhichpassthrough the point are two lines through two points,or == 2. The lines through twopoints may be otherwisefound
by the &Dow!ngconstruction,the tmth of which it ia easy toaee: Draw a plane through the aseamed point 0, and throughthe intersection of its polar planes with respect to the two
quadrics,this plane meets the quadrica in four points whichlie on two right lines interaecting in 0.
A quartic of this species is determined by eight points(Art. 120).
316. Secondiy, let the two quadncs toMh: then (Art. 812)the conestanding on the cnrve has a double edge more thanin the former case, and the developableis of a degree less
by two. Hence
It ought to havebeenatatedaho that the deTetopaMeoircumacribingtwo quadrimhae,M double linM, a conio in eachof the principal planes,seeArt. 108. The numbery 8 !< thM aceoantedfor.
t I owethis jrematk to Mr. Cayley.
CLA68IHCATMNOPCURVES. 265
which expanded contains a as a factor and so reducea to themMt degree. The euepidatedge is the intersectionof <M+3o*,4ce-8<
Since a cone of thé fourth degree cannot have more thanthree doubleedges, two qaadncs cannot touch in more pointsthan ono, nntesa their curve of intersection break up into
simpler curves. If two qnadrics touch at two points on thémme generator, tMa right line ia common to the aortMes,and the mtemect!ombre&ksup into a right line and a caMc.If they touch at two points not on the same generator, theintersection breaks up into two plane conics whoBeplanesintersect in the line joining thé pointa.
817. If a quartio curve be not the intersection of two
quadnca it muet be the partial intersection of a quadric anda cubic. We have already seen that the curve must lie on
a qoadr!c, and if throngh thirteen points on it, and s!x ofherswhich are not in the same plane,*we describea oubic,it muetcontain the given carve. The intersection of this cubic withthé quadric already found muet be the given quartic togetherwith a line of the second degree, and the apparent double
points of the twocurves are connectedby the relation A–&'=='2,as appearaon snbstîta~ng in the formulaof Art. 818the valnea
M=4, <M'<='2,~='8,f='8. When thé line of the seconddegreeis a plane curve (whether conic or two right Unes),we have
~'==0; therefore &=8, or the quartic !s one of the speciesah'eady examinedhaving two apparent double points. It is
easy to see otherwisethat if a cubic and quadric have a planecnrve common,through their remaining intersectiona second
quadric can he drawn for the equations of the quadrio and
cubic are of the form fMe=M,,~='M,a', which intersect on
c,=a!to. If, however, the cubic and quadric have commontwo right lines not in the same plane, this is a system havingone apparent double point, since through any point can be
TMaBmitaRonia neeeMaty,otherwtMtheeaMomightcoMistofthequadrloandofa plane. ThusIfaourveofthéMthorderlieina quadrioitcannotbepMvedthata cubiodistantfromthé quadriocancontainthé
givenearve) MeCMxM~andDublinJMe<XMM<tM<J<?<nM~VoLv.,p.37.
CLASStFÏCATÏON0FCtJKViES.2S6
drawn a tranavemal meeting both Ënes. Smce then X'–t,~=88 or these quartics hâve three apparent double points, and
are therofore essentially distinct from those already discussed
whichcannot have more than two. The numerical character-
!stica of these curvea are precMely the eame as those of the
firat speclea in Art. 316, thé cone standing on either cnrva
having three double edges, and the difference being that one
of the double edgea in one case proceeds from an actual double
point while in the other they aU proceed from apparent double
points.This system of quartica is the rectprocal of that given by
the envelope of o<*+4&~+6c<'+4<&+e. Moreover, this latter
system has, in addition to its cusptdal cnrve of the axth
order, a nodal curve of the fourth which ia of the kind now
treated of.
It is proved, as in Art. SOS, that these quartics are met
in three points by all the generatora of the quadric on which
they lie, wMch are of the same aystem as the lines commonto the cnMo and quadric, and are met once by thé generatoraof the opposite aystem. The cone standing on the cnrve,whose vertex is any point of It Mthen a cnbic having a double
edge, that double edge being one of the generatora passing
through the vertex of the quadric which contains the curve.
Thus while -any cnbic may be the projection of the inter-
section of two quadrics, quartica of this second &mily can
only be projoeted into- cubics having a double point. The
quadrio may be considered as the surface generated by all
the "Knea through three points" of the curve. It is plainfrom what has been stated, that ece~ gMa~c, AactMythree
apparent double ~W)tt, may considered as the M<iM*Me<M~t
of <t quadric tCt<~a cone of <~ <&M-Jorder having one 0~ <~
~e!MM)<fM</ tke ~N<!<M!as <!double e<~e.
818. Mr. C&yleybas remarked that it ia possible to de-scribe through cight points a quartic of this second family.We want to describe through the eight points a côneof théthird degree having its vertex at one of them, and havinga double edge, which edge abaUbe a generator of a quadrio
Ct.ASaïnCATM~0F CURYE8. 367
8
throagh the eight pointa. Now it wu proved (Art. 816)thatif a systemof qaadnca be describedthrough eight points tdtthe generators at any one of them lie on a cone of the third
degree, which passes through the quartic carve of thé Crst
family determinedby the eight points. Farther, if j8",S"be three cubical conea having a commonvertex and passingthrough seven other points, XjS~~S'+fjS" is the generalequation of a cone MaUing the sameconditions;and if it have
Eliminatingthen f betweenthe threodtnereatMf~thelocasof doubleedgeaia thé coneof the 8)xthorder
The intersectionthen of thiacône of the s!xth degreewith
the other of the third determines right lines, through any ofwhich can be describeda quadric and a cubic cone fcMIHngthe given conditions. It is to be ohserved,'however, that théUnesconnectingthe assumedwrtex with the sevenother pointsare simple edges on one of these cônesand double edges onthe other, and thèse (equivalentto jbnrteen intersections)areirrelevant to the solutionof the problem. Four gM<t~Mt<!J~e'
fore canbe<&<eWM<~OM~A~~o&t~.
819. There ia no dinicolty in carrying on this enumera.
tion to curves of higher orders. The reader will find, in theMemoir a!ready cited, a cksmncation of curves of the fifth
order, whîch consist cf three familieshaving four, nve, or six
apparent doublepomts; thé ncetof whichmay have in additionone or two, and the second one, actual double or caspidalpoints. We shati condnde this seettonby applying some ofthe résulta aLfeadyobtained in it, to the solutionof a problemwhich oecasionaUyprésenta M~ "Three sariaces whose
degrees are ~t, p have a certain.carve commonto all three;how many of their points of intersection are absorbed
by the carve? In other words, in how many points do théM)r&cesintersect in addition to this commoncurve?" Nowlot the fint two snr&ces intersect m tho given curve, whose
2B8 CLA98ÏFÏOATMNOfCCBV~
degree ta m, and in a comptementarycorve ~-M, then the
pointe of intersection not on thé nret curve muet be iacladeetin the (~f–M) p intersections of the latter curvo with thethird surface. But some of thèse intersectionsare on thé
cnrve m, since it was proved (Art. 314) that the latter carveintemects the complementarycurve in M(j!t+f-2)–f points
Dedncting this number from (~ff-m) p we find that tho sur-
&ces intersect in /Nt(/t+f+~-3)+f r points which are
mot on the curve m; or that the common curve absorbe
M(/t+~+p-2)-)* points of intersection.In precisely the same way we solve the corresponding
question if the commoneurvo be a doublecarve on thé sur-facep. We havethen to subtract from tho number(~ <M)p,2 {m(/* +v 8) f} points,and we findthat the commoncurved!m!niahesthe intersectionsby w (p-t 8/~+2<' 4) 2fpoints.
These nambeMexpressed in terms of the apparent double
points of the carve m are
820. The last article enables us to answer the question:If the intersectionof two surfacesis in part a cnrveof order
<Hwhich is a double cnrve on one of the surfaces; m how
many points doesit meet the complementarycurve of inter-section ?" Thus, in the exampte last considered,the surfaces
~t,p intersect in a double curvo Mand a complementarycurve
~tp–SNt; and the pointa of intersectionof the three sur&cesare got by snbtrMt!ng from (p.p- 2m)v the number of inter-sectionsof the doublecaTvewith the complementary. Hence
We can verify this formula when the carveMis the complèteintersection of two snr&ces r whose d~reea are k andThen p ia of the form ~!7*+JBPT~ CF* where A is of the
degree p 2&,&c., and is of the form j&~+~F whare Dis of the degree ~–&. The mtemectioNSof the double carvowith thé complementaryare the points for which one of the
tangent planes to one surface at a point on the double carve
XUM'BOJECTÏVE FBOPERTIESOF CCRVE9. 259
82
co!nc!dowith the tMtgeot plane to thé other surface. -Theyare theréforethe intersectionof thc curve with tho surface
~E* J?FjE+ CD*wMch M of the degree p + 8~ 8 +The numberof intersectionsMN {p+2/t 2 (A+~))which coin-cides with tho fonnula already obtained on putting M=M,
~(&+!-8)=)-.
821. From the precedingarticle wo can show how, when
two suriacespartially intersect in a curvc which is a double
cnrve on one of them, thé Btugataritipsof thM eurve and its
compicmentaryarc connected. T!ie Urst equation of Art. 3!4
ceases to be applicablebecauset!io6m'faco/t+)'-2 2 altogcthercontainsthe doublecurvc,but thé secondequationgives us
In Hkcmanner we find that the apparent double points of
the two earvesarc connectedby thc relation
Thus wlien a quadric passes through &double Une on a oibic
the remaming intersection is of thé fourth degree, .of the s!xth
rank, and bas three apparent double points.
SECTIONIII. NOX-PROJECriVEPROPERTtESOF CURVES.
822. As we shaU more than once in this section bave
occasion to considcr lines mdc8u!tcly close to cach other, it
is convonicnt to commence hy s)Mw!ug itow somo of the
formata: obtained in thé Ëmt chaptcr arc modMed when the
lines coumdered are indefinitely near. Wc proved (Art. 14)that the angle of inclination of two lines is given by the
jbnnnh
260 NON-PSOJECHVEPBOPERnES0FCORVES.
It wu proved (Art. 15) that cos~3cosy'-oo8~'cosY, &c.
are proportionalto thé direction-cosinesof the perpendicnlarto the plane of the two lines. It followsthen that the direc-Uon-cosinesof the perpendicularto thé planeof thé consécutiveUnes just consideredare proportionalto M&t-M&M)KM-?&?&M–)tte~thé commondiviserbemg t~M.
Agtua, it waa proved (Art. 48) that the direction-cosinesofthe line bi&ectmgthe obtnse angle made with each other bytwo Uneeare proportionalto
co8tt–cos<['}cosj8–co9~ cos'y–cos'y',&c.
Hence whentwo linesare indefinitelynear, the direction-cosmosof a line drawn in their plane, and perpendicular to theircommon direction are proportional to 8 cosa, 8 coB~, 8 cosy,the commondIviBorbeing M.
323. We proved (Art. 286) that the direotion-coainesof
a tangent to<&! <)~ <&
1.< «. hoa tangent to a curvoarey whue, n thé carve be
given aa the intersection of two surfaces, these cosines are
proportional to ~y-~f'j~ ~Z'Z, Z~Z'J~ where
L, &c. dénote the arst dif~rential coeScients.An infinity of normal lines can evidently be drawn at any
point of the cnrve. Of thèse two have been distinguishedbyspecial names; v!z. the normal which lies in the osculatingplane which is commonly oalled the ~nac~x~ KM'BM!J'/andthe normal perpendicular to that plane, which being normal
NON.PBOJBCnVBMOMENTIES0F CUBVM. Ml
to two consecattvedémonta of the carra bas been c<JM byM. SMBt-Venant&6 Binormal.
ARthe nonnab lie in thé planeperpendicoha*to the tangentline, ~Iz.
824. Let œ considernow the equation of the oscalatmgplane. Sinceit containstwoconsecutivetangents of the curve,ita direction-comnes(Art.822)are proportionalto
quantitieswhichfor brevi~ weah&Ucall X, Y, Z. The eqt)<~tion,of theosculatingplaneis therefore
The same equationmight have been obtained (by Art. 80)by forming the equation of the plane joining tho three con-secutivepoints
In applying this formulawe may simplify It by taking oneof the co-ordinates at pleasnreaa the mdependent variable,mdMmaHng<iFa!,<yot<<=!0.
826. In order to be able to iMostrateby an example the
applicationof thé &nm)Iœof this section, it îa convenienthereto form the eqattiom:tmd state Mme of the properties of theAe?N!or curve &nnedby the threadof a acrew. The helix maybe de&ted as the form aasamedby a right line traced in any
plane when that plane ia wrappedround the surfaceof a right
cylinder.* From this definitionthe équations of the he!ix are
ConveMetyahelixtecomeaa rightUnewhenthe cyttatteron whichit ïa ttMedh developedintoa plane,andis therefoMa géodésieon thecylinder(Art.278).
882 NON-MOJECTIYBPBOPEBTÏE9OP CURYES.
easily obtained. The equation of any right Hne~'=Mt!B ex-
presses that thé ordinato is proportional to the intercept which
that <mluiatc makes on the axis of x. If now the plane of
the right line be wrapped round a right cylinder so that the
axis of a: may coincide with the circular base, the nght lino
will heeome a belix, and thé orAinato of any point of thé
curvo will be proportioual to the intercept, mcasurod alongthe cMc, tvhidt thilt ordinate makes on thé circular base,
counting from suy nxed point on it. Thua the co-ordinatca
of thc projection on the ptane of thé base, of any point of
tho helix are of thé fbnn a:==<itco' y=.o8tn0, where a is
thé radins ot' the circular base. But thé hcight e has been
just proveù to bo proportional to thé arc 6. Ilence the eqna-tions of the helix aro
We plainly get the same ra~cs for x and y when the arc in-
cn-ascs by 2?-, or when increases by 2rh hence tho interval
between thé thi'cads of thé screw is 2~.
Since wc have
the angle made by thé tangent to tho helix with thé ax!aof (whieh is tho direction of thc gencrators of thé cylinder)H constant. It !a easy to sco that thia !a the same aa thé
angle made with the gouerators hy the line into wMch théheltx is developed when the cylinder is devetoped into a
plane.The length of thé arc of tho cnrvois evidentlyin a constant
ratio to the height ascended.
The cqaattONiioftho tangent aie (Art. 286)
If then a: and y be thé co-ordina.teaof the point whcre the
NON.MtOtECnVE MOPERnES 0F CUBVM. 268
or the distance betweenthe foot of thc tangent and thé pro-jection of the point of contact ia equal to the arc whichmeaaarea tho distancealong the circle of that pro}ectMnfi'omthe initial point. This also eaa be proved geometrically,forif weimagine the cylinderdevelopedout on the tangent plane,the helix will coincidewith the tangent line, and the line
joining the foot of thé tangent to tho projection of the pointof contact will be the at'o of the circle developedinto a rightline. Thns them thé locus of thé points whare the tangentmeets tho haM is the involuteof the circle.
The equation ofthe normalplane is
The &fm of thoequationshowsthat thé osculatingplane makea
a constant angle with the plane of the base. We leave it
as an exercise to the reader to find thé tangent, normal
plane, and osculatingplane of thé intersection of two central
quaJnca.
826. We caa give thé equation of the osculating planea form more convenientin practice when the curvo is givenas the intersection of two surfacesU, K Since the osculating
plane passes through the tangent line, îts équation muet be
of the form
where 7/.C+&C.ia the ttmgent plane to thé ËMtsurface. This
eq~tton is identicallyMÛe&edby the co~)rd!natcaof a point
264 NOÏt.PBMECTITE MOPEMtES 0F CURVE8.
eonunon to thé two surfaces, and by those of a consecutive
point; end on sabsëtating the eo~rdhMttesof a secondcon-
secutivepoint, we get
and smce we may either, as in ordinary Cartes!an equations,take w M constant; or dae <e,or y, or e; or more geaeraHy
may take any tmear fonctionof theMco-ordtaatesas constant;we may therefore add to the two preceding equations the
arbitrary équation
NtwthiadetennmanttN&ybereduoedbysabtracting&omthe6Kh cotmnn mt)lttpl!edby (m–t) the sma of thé &'at four
colanms,maHipUedrespectivdy by a! !o; when thé wholeof the Sfth cottuan vMuaheaexcept the last row whiehbecomes
NON-MMECnVE PROMKHEB 0F CURVE8. 2M
-(aic+~Sy+'j~+SM). In like manner we may then subtractfrom the Mh row multipliedby (m 1) the anm of the &'stfour rowe multipliedrespecttvetyby a', M,when in likemanner thé whole of the Mh row vanishes except the 6AhcohmmwhichM (aa? + +<y<!+&o). Thaa the deto'nunantreduceato
If we calI the determinantlast written 8 and the correspondiagdoterminant for the other equation the équation of tho
osculating plane is
327. The conditionthat fourpointa should lie in one plane,or in other worda,that a point on thé cnrve ehould be the
point of contact of a Btationaryptane, ia got by Mbstttatmgin the equation of thé plane throngh three conseen~a points,thé coordinateaof a fourth consecutivepoint Thus from the
equationof Art. 324the conditionrequired ia the determinant
ThMéquationh due to M.HeMe, see Cretle't Jb«fM<t<,VoLxn.
266 NON'PBMECnVB PROPMTtES OP CUNYE9.
If a curve in space be a plane curve, this condition mnst
be fulfilled by the co-ordinates of every point of it.*
828. We shall next consider thé circle determined by three
consecutive points of the carve, which, u in plane cm'ves, is
called the circle of curvature. It obviously lies m thé oacu-
lating plane: its centre is thé intersection of the traces on
that plane, by two consecutive normal planes; and its radiusM commonly called the radius of <t&Mhfecurvature, to dis-
tinguish it from the radius of NpAeMca~cnrvature, which ia
the radius of the sphère determined by four consecutive pointson the curve, and which will be investigated presently. If
through the centre of a circle a line bo drawn perpendicularto its plane, any point on this line is equidistant from all the
points of the circle, and may he called & pole of the circle.Now the intersection of two consecutive normal planes, evidently
passes through the centre of the circle of eurvature, and is
perpendicular to its plane. Monge bas therejbre called thé
lines of intersection of two consecutive normal planes, the polarlines of thé surface. It Is evident that all the normal planes en-
velope a developable of which these polar lines are the generatora,and which accordingly has been called thé polar surface. Wc
shall presently state some properties of this surface. The polarlino is evidently parallel to the IIne called the Binormal
(Art. 32S).
329. In order to obtain the radius of curvatare we shall
first calculate thé angle of coK&!C~that is to say, thé anglemade with each other by two consécutive tangents to the
1 have not Mceeeded in completing thé reduction of the correspondingtondMon when the carre is given as the intersection of two eat&ee< P.M.BUcho~ (0'<!<b,Vol. LVin.) gives as thé resulting condition the jMeMan
ofthe four surfaces U, Y, S' (see Art. 16S)) but M. BhchoS's MMonin~le unsound, and hMresult Monly correct in the case where the Mt<ace<
are quadries. The condition in gênerai ia of the degree CM+Cx-ZO
in the eoefMents, as might he inferred &om the value of «,Art. 810. ïtis the sum of two terms, one of which M the Jaeobian, and the othet
is the same function of thé 6Mt and second dift'erential coeCcients as
the JacoMan M when the surfaces are qnadHM.
267NtMf-HMMECTIVEPROPEttMESOP CCttVBS.
curve. Thé direction-cosinesof the tangent being ) y < S;
it follows from At't, 822 that <? the <mg!e between two con-
sécutive tangents is given by either of the 6)rmu!œ
The truth of the latter formulamay bo seen geometrically:for the right-handmdoof the equation denoteathe square of
doublethc triangleformedbythreoconRecuûvepoints(Art.SI) ibut twoaidesof this triangleare each<&,and thé angle betweenthem is dO,hencedoublethé are~ is
If now (&be thé element of the arc, thé tangents at thecxtrem!t!esof which make with each other the angle dO,thendnce the angle made ~!th each other by two tangents to ac!rc!eis equal to the angle that their points cf contactsubtendat its centre, wo hâve p~==<&. And thc élémentof thé areaud thc two tangents be!ng connnon tu the curro and thecirde of ourvature, tho radius of curvatare is given by theformula
By performing thé di&renUatîoM indicated, another value for <?*
M&<mdwithout diNculty,
This formula may <tkobe proved geomet)'!ca!!y. I~et ~J!, BO be two
consecutive elements of the cmTe! ~D a Une parallel and equal to JBP;then emce thé ttrojeettOMof BCon the axes Me <&t+ <<*<dy t d'y, <&+<<*<,it ia pMn that the projections on the axes of the diagonal CD are d'jr,
d'y, <<*<,whence C~' (t!)!f t (<~)' + (<~)*. But CD projeeted on thé
element ofthe arc ia <<*<,and on a line perpendicular to it is <&<?:whenco
NON-HMMEtttVB fBOPEKHES 0F CPBYM.268
Ex. To &td thé Mdiut of CM'Mtt«Mof thé hettx. tMag thé &M'mutme'~A*
ofAtt.SM, weSad~a–t Mdta]f<tdia<ofen)n~tt)''ebMMtMt.
830. Havmg thus detemuned thé magmtnde of thé rad!a9
of omvtttore, we m'e eMNed by the formulo of Art. 822 aïso
to detennuM ita pûaition. For thé ditecdon-coMnes of a line
dMwn m thé plane of two consécutive tangents, and perpen-dicalar to their common direction are hy that arttde,
If Il he the co-ordinate of a point on thé eurve,and x, y~ « those of thé centre of carvatnre, then the projec-tions of the radius of cnrvatttreon the axes are a!a:, y'e'–i!; but they are a!ao p coeet,p cos~S,p cosy. Putting inthen forcoset,cos/8,coa'ytheir valuesjnst found,the co-ordmatesof thé centre ofcorv&tareare determinedby the equations
831. When a curve Mgtven as thé intersection of two
sor&cMwhich eut at right angles, an expresion for the radius
of carvatore can be eamiy obtained. Let and < be théradn of curvature of the normal sections of thé two surfaces,thé aectMnabeing made along the tangent to thé curve; and
let be the angle wMch thé oscola~ng plane makes withthé omt normal phme: then by Mennier'atheorem, we have
The Mme equations determinethé oscat&tmgplaneby tho
&))'mn!at<m~=_y.
If the angle whichthé Mn'&cesmake with oach other be <e,
theconespondiDjg&nntthis
NMHMtMECMVE NtOPBtfFtM OP COKVES. 86&
We can henceobtain an expression for the radius of cnr-~vature of a cnrve given as the interaeotion of two <Mr&ce9.We may write Z'+Jtf+~=J! Z"+Jtjr'+j~=JE"; andwe have
832. Let us now considerthe angle mado with each other
by two conaecativeoscotatmgplanes, which we shall caU the
angle of torsion,and denote by < The direction-cosinesofthe osculatingplanebeing proportionalto JT, 1~ the second
&rmula of Art. 822 gives
NON'MMECTtVEPRÛPEMIE80F CCRTKS.270
Thia fbrmuta may bo also proved geometiieally. For J)/
dénotes six times the volume of the pyramid made by fourconsecutivepoints, whHe~*+y+~* dénotesfour tuucs tho
square of thé area of the triangle formedby throe consecutive
points. New if be the triangular baseof a pyramid,A' an
adjacent face making an angle with the base, Nthe aidecommonto the two faces, &ndp thé perpendicular&om thé
vertex on a, so that 2~1'==<p:then for tho volumeof thé
pyramid we have3~T==.~ain~and 6~!=8~p~ Nn~==4A~'sun).Now in the case considered, the commonside la ds, and m
thé llmit henee 6r& == Q.E. D.
Following thé analogy of thé radius of cutvxtare which ia
thé later Frontiti wntera denote thé qnaBtity*y
by the
letter r, and caR it thé radius of (oMM)!but the reader will
obMT~ethat th~ ia not, like the radius of curvatttre, the radiusof a reat circleinthnately connectedwUathé corve.
333. In the samemanner, howcver, Mwe haveconsideredan osculating circle determined by tbrec consecutivepoints ofthe ayatem,we mayconsideran osculatingright conedetennmed
by three con~eut~c phnea of t~o aystem. Imagine that a
ephMeM descnbed having as centre tho point of thé aystemin wbtchthe three planes intersect; let thé Unesof the system
The quaatityyis also Mmetune~calledthe "Meondcntmtmfe"of
the<Mtvo.
NON.PMJEC'nVE PBOPEBTÏES0F CURVES. 271
passing through that point meet thé sphère in jl end JS;and let thé correspondingplanes meet the same apherem
BT; thenifwe describoa small circle of the same spherepassingthrough and J3, and toudied by -~7~ BT, thé conewhose vertex is thé centre, and which stands on that smallcircle will evidentlyoseulate the given curve. Thé problemthen is, being given the angle between two consecutive
tangents to a mall circleof a aphere, and <?~the correspcndingarc ofthe cu'cteto findJ~Its radius.
Let C be the centre of the circle, and from thé right-
angled triangle CAT wo hâve 8m~2'=. If thentan~~U
be the externat angle between two tangents to a oirde,< the length of the two tangents; H the radius of thé circle
!s given by thé formula tanir=~ In thé limit e is
<f~thé element of the arc of thé cirdc, and tan.B=, or ac-
cording to thé notationused, tanB'c' --=-du p
334. Imagine that through every line of the systemthereis drawn a plane porpendieularto the correspondingosculatingplane, the assemblageof these planes generatea a developablewhich is called thé ''<c< developable. Thé reason of théname is, that the given curre is obviousiya gcodesicon this
developable,sinccits osculatingplane Is, by construction,everywhere normalto thé surface. If thereforethe developablebo
developedinto a plane, the given curvo will become a rightline.
The intersectionof two consecutiveplanes of thé rectifyingdevelopableis thé }'cc<~<~/< Now since thé plane passingthrongh the edge of a right cone perpendicular to its tangentplane passes through its axis, it follows that the rectifyingplanepassesthroughthe axis of the osculating cone considered
It has beenprovedty M. Bertrandthfttwhenthe ratio f p M.coMtMt,thecurremustbe a holixtMeedon a cylinderandbyPuiseux,thatwhenr andp arebothconstant,thecylinderhn a circutarbMc.
NON-HMMECnVE MOFBBTTB! OP CCBVES.272
in tho last article; and therefore that the Mc<~«!~ line is
the OMMof <&o!<<woM~!<Mtyc<MM.The rectifying line may be
therefore construeted by drawing in the rectifying plane a
line making with the tangent line an angle wheroF hasthe value determinedin the tast article.
The rectifyingsar&tceis the surfaceofeentreaof the original
developable. In fact it was proved (Art.277) that the normal
planes to the original Ntr&cealong the two principal tangentstouch the surface of centres; but the generating line hsdfis in everypoint of it one of the principaltangents; the recti-
f~ing plane therefore touches the surface of centrea 'wHch is
the envelope of al! these rectitying planes. The centre of
carvatore at any point on a developableof the other principal
section,namely, that perpendicolar to the generating une, is
the point where its plane meets the correspondingrectifyingline, for evidently the tracea on this plane of two consecutive
rectifying planes are two consecutivenormalsto the section.
Bence if he the distance of any point on the developablefrom the cuspidaledge measuredalong thé generator,the radiusof corvatnM of the transverse section is Man.B. When 1
vanishes, fhis radius of curvature vanishes as it ought, the
point being a cnsp.In the case of the helix the rectifyingsurfaceH obviously
the cylinderon which the curve is traced.
338. 'M? <~OK~ &e<!OMM<WtMCCeMtWM!d'tt<<<fM!<Mr&Let AB, JBuM traces on any
spherewithradius umty, of planea
parallel to the osculating and
normal planes, then the central
radius to B iathe direction of thé
radius ofcnrvature. If ~J3', B' C
be consecutivepositionsof the os-
culating and normal planes, B' is in the directionof the con-secutive radius of cnrvatnre, and BB' meaMMSthe anglebetween them. Now the triangle BOB' being a vay smaN
right-angled triangle, we hâve
NON-HMMBcnVEPMOPËKTIES0F CURVE8. 278
T
Bat emeethe angleABC m right, ~<3mcasnresBAB', which
ia < the angle betweentwo consecutive osculating planes,and OB' measares OCB', whieh is dO, thé angle betweentwo consecutivenormalplanes. The required angle M there-fore given by the formula ~B"==<+~; where <&)and
<? have the values already found. The series of radii of
curvatnre at all the points of a curve generate a surface on
the propertiesof which we have not space to dwdl. It is
evidently a skev surface(seenote, p. 76), since two consecutiveradii do not in generalintersect (seeArt. 338,infra).
Ex.1. Tofindthe equationof the sm'&ceof theradii of curvaturein thecaseof thehelix.
Theradiuset cutvattitebeingthé tnteKMtionofthe OMuMngandnormalplsneahasfor !t<équations(Art.326) ~):, <!° &Mawhichwearete eUmittoteit~t'bythehelpof thé equat!oMof the eurve. Andwritingtheequationsofthehélixit=o co<M,y a Binoe,the requiredmt&eeieyeosttf.tetaM!.
JEt.2. ToHndthe équationof the developablegeneratedby the tan-gentsof a helbt. Theequationsof thé tangentbeing
(< «CM<t<')f -oa tinM*(<-<'), y e e!nM's <Mtcoa~ (e <'),theMsaltof eliminating<iefoundtobe
SincethisequationbecomeaImpossiblewhena;*+~*«! it MpMnthatnopartofthefHM&oelieswithinthecyUnderonwhiohthehelixistraced.
836. We shallnowspeakof the polar developablegenerated
by the normal planes to thé given curve. Fonner hM re-
marked, that the "angle of torsion" of the one system !a
equal to the "angle of contact" of thé other, as is SMJBcienttyobvtOQSsince the planesof this new system are perpendicularto the Unesof the originalsystem, and vicew<!< The reader
/Mwill observe howeverthat it doos not followthat the of
<&j~one system ie equal to the of the other, becauao the <&is
not the same for both.Since the intersectionof thé normal planes at two con-
«ccuttvcpoints J?, JT' of thé eurve is tho axis of a circle of
NOtt-PROJECTïVBFMPNMtMor CPKVE8.~4
tvhich9 aud Jr &Mpoints (Art.828), it followsthat if anypoint 1) on that line he joinedto and K', the joining Imc~are equal and make oqual angles with that axia.
It M plain that three consecutivenormal planes mteraectin the centre of the osculatingsphère hencethé CM~px&t~edgeof <J~polar <&M?Opa5~M locusof centresof ~eMCO~CM~-vature.
In the case of a plane corve this polar developablereduceeto a cylinderstandingon the evoluteof the eurve.
337. J~t~ty curve ~<Man t'H/&t~ of e~M<M~Mtyon the
~o?<M'<~M!<~MtMe/*that is to say, the ghren carve may be
generated in an infinity of ways by the unrollingof a stnngwoundround a curve traced on that developable. Let 3)Hf',~f'Jtjf",&c.denote thé successiveelementsof the curve, Jï',&c<the m!ddlopoints of thèse éléments,then the planes drawn
through the points ir pcrpendicularto the elements are thénormal planes. The l!nes AB, ~l'J3', &c. are the lines inwhich eaoh normal plane is intersected by the consecutive;thèse lines bemg the generators cf the polar developable,and
hencetangentsto the cuspidaledgoJ!~ of that Mr&ce. Draw
now at pteasutef any tine JH) in thé 6mt normal plane,
meeting the first generator in D; joïn DJT which being in
the secondnormal plane will meet thé secondgenerator~'B',
say in D'. In like manner, lot J8r"D'meetA"B" in 2)". We
SeeMong~p.aM.t TM<C~areiat<Aen&omLeroy's6<M))M~y~fMrM2)MtMo<Mtx.
NOX.PBMECTIVEMtOPERTtH)Or CCNVE8. 276
f2
get thus a curveDD'D" traced on the polar developablewhichie an evoluteof thé given cnrve. For the lines2)jS, jy~, &e.the tangentsto the cnrve 2)2)*j?", are nonnak to the carve
-KX~ and the lengths JP~~D~ jP'jr=.Z)'JP",&c. (seoAtt 886). If thereforeDK be a part of a thread woundround
.MM)", it is plainthat as thé thread !s unwoundthe point Kwill movealongthe givencurve.
Since the firet line DK was arbitrary the carve has an
infinity of evolutes. A plane curve bas th)B an infinity ofevoluteslyingon the cylinderwhosebase is thé evolute in the
plane of the cnrve. For example, in the special case wherethe evolute Kdncesto a pomt; that M,when the eurve is a
éircle, the circle can be described by moving round a threadof constantlength fastenedto any point on the axis passingthrough the centre of the cMe.
In the general case, aU Me et~«<ecurves D2)'jP", <&e.are
~eo<&<Meon<~j)o! d5K)d~p<!&&.For we have seen (p.819) that a cnrve is a geodesicwhen
two snceesmvetangentsto it make equal angles with the inter-section of the correspondingtangent planes of the surfaceand it hae just been proved (Art. 886) that .M, DK whichare two successivetangents to the evolute make equal angleswith J[B whichis the intersectionof two consecutivetangentplanes of the developable. An evointe may then be found
by drawinga thread as tangent from K to the polar develop-able, and windingthe continuationof that tangent &eely roundthe developable.
338. The locnaof centres of curvature is a curve on the
polar developable,but M not one of the eystem of evotntes.
Let the nret oscu1atingplaneMM'M" meet the first two normal
planes in Kg ~'C, then C is the arst centre of cnrvatore:
and in like mannerthe secondcentre !s C', the point of !ntep-section of JT'C",J5'"C', the Unes in whieh the second oscu-
lating plane3f'Jtf"J(f" inmet by the secondand third normal
planee. Now thé rad!! JT'<~~'C" are distinct, since theyare the intersectionsof thé same normal plane by two dînèrent
osculatingplanes,2<T'C'will therefore meet thé lino ~2! in a
NON-PRMECHYEPBOPEBTÏESOf CURVES.276
point J which is distinct from C. Consequently thé two radii
of cnrvatnre J8'C~ ~T'C" situated in the planes jF~F* bave no
common point in ~J? thé intersection of these planee; two
consecutive radii therefore do not intersect, anteas in thc case
where two consecutive oscalating planes coincide.
The centres of curvature then not being given by the sao-
cesMve intersections of consecutive radu; these rttdH are not
tangents to thé iocus of centres. Any radius therefore JSTC'
would not be the continuation of a thread wound round CC'O",and thé unwinding of sach a thread would not give the curve
jnr' except in the case whero thé tattor ia a plane curve.*
839. To ~M<~the f<tJtM of Me ~?8 <&OMy&J~MM'OOM-
secutive ~(M'H~. Let bo the radius of any sphere, p the
radius of a section by a plane making an angle with thé
normal plane at any point; then, by Meunier's theorem,
~cos~t=p; and for a consécutive plane making an angle
We have then oniy to g!vo in th!s osprMSton to p and <<~thé values aiready found (Arts. 330, 382).
m obvïousiy t!to length of thé perpendicnlar distance
from thé centre of tho ephere to the plane of the <MTdoof
cttrvatufe.
840. 2~ j~K<~the co-m~MM!<M ~e ceM<n:of tXe oscM&t<M!~
sp~~e.Let tho equation of any nonnt~ ptanc be
where a~ !s tho point on thé curve, and a;8'y any point on
thé plane; then thc cquation of a consécutive normal planecombined with the prcceding gtvca
Thé chMMterMtiesof thépolar developablemay be investigated byatgomen<s8imilar to thMe umd Higher ~fMe C«rMe,Art. tl6! thus itia easyto see that the etaM of that developableM mf. whero m and rhavetheMme meMun~M et p.234.
NON-PNMECnVBrfM'FMM'tESOf CURVfM. 277
By squaring and adding theso équations we obtain another
expression for JB*,which !s what the value in the last article
would become when for p acd we substitute their values.
We add a few othcr expressions, thé greater part of which
admit of simple gcometrical proofs, the detaib of wMch want
of spaceobliges ns to omit.
Ex. Ï. If <rbe the Mo of the eurve whieh fa the tocuoof eentKWofabeoluteouMatme,
Ex. 2. If Sbethe length of the arc of the locusof oenttesof ephetîeat«u'vatare <? = where& is the distancebetweenthe centresof
e a'!the oMMiatingeircte and (Mcahtitt~ aphere. From thie expression we
immediately get values for the radii of carvature and of torsion of thia
béas, remembering that the angle of torsion M the angle of contact of
the original and vice veM&.
Ex. 8. The anglebetween two conaecative reotifyng lines la dB.
Ex. 4. The mgle between two succeMiveJ!'a i$givet~by thé fonnuta
The Kadetwittûnd Attther deta!h on the eubjectf)treated of in this
section in a Memoir by M. de Saint-Venant, .Tbtotxt~<h r~!eo&jPo~<M&-<t<{W, 0<MM'XBC.,who has also collected into a table about a hundred
&Ktndte for the transformation and reduction of ealculatione relative tothe theory of mn-phme e))jve<) and in a paper by M. Frenet, LioaviUe,Vol. xvn., p.4!7. 1 abridge the foUo~ing hi<tortcat sketch &om M. deMnt-VeMn~a Memo!r Cnrvelines not contained m the 8MMp)ane havebeen encceasivelyetudied hyClairaut (~<eA<M~««<)' &< <M<f&Mà <<M<Ne
<M<f6M'<,n31), who bas brought into use the titlo by wh!eh they hâvebeea commonlyknown (previously, howe~er, employed by Pitot) and who
CtfBVEi!TBACED ON SURFACES.278
SECTION IV. CURVE8 TBACEDOH SURFACES.
341. It remains to say somethmgof the propertiesof cuvesconsideredas belonging to a particular sm'fye. Thue the
aphereve know bas a geometry of ita owa, where great circles
take the place of ImMin a plane; and in like manner eachsarfacehas a geometryof its owa, thé geodeNMon that surface
anewenDgto right lines.
We have ahfeadyby anticipation given the fandmental
property of a géodésie(Art. 978). The dfferentia equation)a Immediatelyobtainedfrom the property them proved, thatthe normallies m the plane of two sacceaatveelementaof the
carvo aad bMectethe angle between them; hence Z, J~ N
whieh are proportionalto the duection-cosmettof the normal
direction-cosinesof thé bisector (Art. 822). Thns "if the tan-
gente to a geodesicmake a constant angle with a fixed line,the normalealongit will be parallel to a fised plane, and WMMM<!(Dickson, C!M~~t%eand Dublin ~M!!<Xe<Ma<M<t?JoMraa~Vol. V.,p. 168). For fromthe equation
bu given expressionsfor thé prqjecttoM of thèse carm, for their tangents,nMmah, arc, &c. hy Monge (3SmoMWt«r le, <KM&<)~<,~< pteoentedin 1771, and hMerted in VoL x., 17M, of the '&e<M« ~ax~t,' Mwell M in hh 'p<«M<Mtt <<t<Ma~M tt G&tmarM') who gave ex.
preadoM for the normal plm, centre and radius of eun~tuM, evolutes,polar lines and polar devdepaMe, centre of osculating sphère, for theontenon ibt 'pointe of ~mpk înBMtion'where four eoMeenttve pe!n~ arein a plane, and &)f <pohn<ef double MexMn' where three coaMM~ve
pointa are înatightUne! by'NMean(&<<t«Mt<b9«<~tt«~<~M~NMt,&e.presented in 1774, jStMH<t~M~ft, Vol. !x., 1780) who wM the nMtto eoMMee the osculating plane and the developable generated by the
tangentai by Laofoh (OtM .D~rot~M) who wae the nMt to Kndeîthe fonnutee tymmetnoal by introducing thé diCeteatiah of the three
co-ordiMtea; and by Laneret (JM~noMWMffj~ «xo~M d!Mt6& <!MO~«M,read 1802, and inserted VoLI., 18M, of &t<XM<tétranger, de l'Inetitut)who cahalated the angle of torsion, and !atMd)Medthe consideration ofthe KctHy!n~ lines and Kctnying surface."
CURVE8TRACEDON 6UBFACJM. 279
which dénotesthat the tangents make a constant angle witha Ëxad line, we eu (Mnc~
whichdénotesthat the notmats are parallelto a fixedplane.
842. If ~M«~ any point on a surface ~e~ <~<Mt<tM
~M&~KM~a~~M~~M~MM~?Me~ww~<& ez-
<yem<MMta at )'~< aa~ to to<&Let ~LB=*~<7and let us suppose the angle at B not to
be right, but to be =8. TakeBD=- e
and then becauaeall the aides of the tri-
angle BOD are in&utely am&Hit may b<ttreated M a plane tnangle and the angleDCB is a right angle. Wo have therefore
DC<2)~, jiD+DO<A0, and therefore<~<X It follows that .~<7 îa not theshortest path from ~Lto C, contrary to hypothema. Ur the
proof may be stated thus Thé shortest line &oma point JL
to any curve on a surface meets that carve perpendicu1arly.For if not, take a point D on the radius vector &om and
indefinitelynear to the carve; and from this point let &!1a perpendicularon the curve [wh!ch we caa do by ta!dngalongBU a portion=BD cos and joining the point so foundto D]. We can paM then from D to the curve more shortlyby goingalongthe perpendicular than by travelling alongtheasaumedradiusvectorwhichis thereforenot the ehortestpath.
Henee, if every géodésie through A meet the curve per-
pendicularly,the length of that géodésie ia constant. It 18
aiso evident mechanicaBy that the ctrcte descnM on anyNU&coby a sttamed cord from a 6xed point is every where
perpendicularto the direction of the cord.
343. The theoremjast proved ia the &tndamentaitheoremof the methodof m&MtesimaIa,applied to right lines (C!M!t<y,
fhMtheoremis duetoGaum,whoaltopMVMit bytheCalculMof
V<niationt)<eetheAppendixto Li<mvUle'<EditionofMange,p.628.
CUBYESTBACEDOM 8UBFACM.280
pp. 289, &c.). Ait the theorems therefore which are there
proved by mcans of this principlewill be tme if instead of
right lines we cons!dcrgeodesicstraccd on any suriace. For
example, if wo construct on any surface thé carre answeringto an ellipse or hyperbola; that ia to say, the locusof a pointthe atttn or (USereaooof whose geodesie d!ataïMeafrom twoSxed points on the surface is constant; then the tangent at
any point of the !oensh!sect8thé angle betweenthe geodesicsjoining the point of contactto the 6xod pointa." The conveneof thistheorem is alsotrae. Aga!n, if two géodésie tangentsto a cnrve, thmugh any point P, make equal angles with the
tangent to a curve along which P moves, then the diferencebetweenthé snm of these tangents and the intercepted apo ofthe curve which they touch ia constant" (see (~MMM,Art. 856).
Again, "if equal portionsbe taken on the géodésie normabto a onrve, the line joining their extrcmities onts all at rightangles," or "if two different curreBboth cut at ïight anglesa system of geodesicsthey intercepta constant length on eachvectorof the series." We shallpresentlyapply these principlesto the case ofgeodesicstraced on qnadrica.
344. As the cnrvatureof a planecurve is moasuredby theratio which the angle betweentwo consecutivetangents bearsto the element of the arc; so the ~ax~M cM~x!<MMof a carveon a sarface Is measaredby thé ratio borne to the élémentof the arc by the angle betweontwo conseoativogeodesic
tangents. The followingcalculationof the radius of géodésie
carvatare, due to M.HonviRe,* gives at the sametnne a proofof Meanier's theorem.
Let mn, Mp be two consecutiveand equal elements of the
curve. Produce nt==N!K,and let Ml
the perpendtcnlar on tho plane
mnp. If now be tho angle of con-
tact <p=6<&. Now flq is tho second
element of tho normal section: let
tnq= 6',then is the angloof contact
Appendix to Monge, p. 676.
CCBVE8TRACEDONSURFACES. 281
of the normal scet!on,and <j'==6'<&.Now thé angle g<p(==~)Mthe angle between the oseulating plane of the eurve andthé plane of normal section, and since <g==<pcos~we have
<=~cos~ and n =* whieh laMeunter'a theorem; R being
théradius of cnrvatare of thé normal section andp that of the
givencurve.
Now, in liko manner, jpMgbcing 6" the géodésieangle of
contact, we have pq 0"de and pq ip sino,J. or
1àino.contact, we h&ve ~c!0"<& and ~=<pa!a~, or -==–
The geodeeM~radius of curvature is thepe~re It isdino
easy to see that this geodeNLCradius !e the absoluteradius of
~furvatureof the plane cnrve into which the givencnrve wouldbo traosfbnned,by ciroumscnbmga developableto the givensurfacealong thé given curvc, and unfolding that developableintoa plane.
345. The theory of geodesics tracod on qaadncs may be
said to depend on Joachunsthars fundamental theorem thatat every point on snch a curvc pD 1s constant where, as atArt. 174, p is the perpendieularon the tangent plane at the
point,and D is the diameterof the qnadric paraUetto the tan-
gent to the cnrve at the samc point. This may be provedby thé help of thé two followingprhiciples: (1) If from anypoint two tangent lines lie drawn to a quadric, their lengtheare proportional to the parallel diameters. This is evidentfrom Art. 70; and (2) If from each of two points ~1,B onthe quadric perpeudicularsbe let fat! on the tangent plane attheother, thèse perpendlca!arswill be proportionalto the per-pend!onlarsfrom the centre on the same planes. For the
length of thé perpendtcnlarfrom a:'y~" on the tangent plane
1have not adopted the name second géodésie em~tare" !ntrodu<ed
by M. Bonnet. It is intended to expteM the ratio borne to the élémentof the arc by the angle which the nonnal at one extremity makes with the
planecontaining the clement and Uie normal et the otho- extremity.
CUBVE8 T&ACEDON SURFACES.2M
If now from thé points B there be drawn UnesJ[ BT
to any point T on the intersection of the tangent planes <t~i and B, and if AT make an angle i with thé intersectionof the ptane~ the angle betweenthe planesbeing w then the
perpend!cnlar from A to the intersection of the planes is
~3"mNt and from A on the other plane js ~jfsmtNnM.In like manner the perpendîcular&om on the tangent planeat AisBT mn<'sm~. If now the lines BT make equalangles with the intersection of the planes, the linea BT
are proportionalto the perpendiculars6'om and Bon the tw~planes. But J[T and BT are proportionalto D and 2)*,and
the perpendiculars are as the perpendïcnlarsfrom the centre
and p. Hence 2~==JP'p'. But It wu proved (Art. 878)that if ~4 TB be saccessiveelementsof a géodésiethey make
eqnal angles with thé intersection of tbe tangent planes atA and B. Hence the quantity pD temaiMunchanged as we
pass from point to point of the geodesic. Q.&&
846. On accountof the importanceofthe precedingtheoremwe wish atso to show how it may be deducedfrom the dine-
rential équationsof a geodesic.f DMerentiatingthe equation
TMaproof is by Dr. Chfavef),Ct'<N<,VoLxui., p. 279.
t SM~oMMtMthaï, C~Nt.Vot.xxvr., p. IM; Bonnet, J!)«nt«<<&f~M&
J'ei~tetaj~, VoL xix., p. t88t Dickson, Ch~M~ and Dublin JMa<
<H<t<Mj~Ji)Mr~M~VoL v., p. 168.
CURVMMACEBON&UBFACE8. 288
It is to be remarked that thia equation ie (Jao tme for a
Uneof eurvature; for Bincop,
&c. are the d!mct!M~*coMnesof
the normal, the d!rectK)n-coame&of a l!ne in the same planewith two consecattvenormale and pefpendtcular to them are
(Art. 822) proportionalto <f (p)&c. Heace the
-y-,&c. of
a Uneof curvature are proportional to~(~).
But if now
wc diffarentiate
If weactuallyperformthe dMerentiatuMM,&ndreduce the remit
by the diferential equationof the aarSMeJMc +~Mi~')-~Hk= 0~and !t~ conséquence
347. The:precedmg equation is true for a geodeac or Uneof carv&tnMon any MU'&ce,but when the surface.ia'~ty of
the Meonddegree, a &mtmtegr&lof thé equation can be found.In fact wehave
TMa may be easily venËed by namg thé general equation ofa qaadno, or more simplyby osmg thé equation
CURVES TRACED ON SURFACES.284
by sabstitntmg whiehvalues the equationis at once established.The equation of the last article then consistaof tenoBeach
aepantely integrable. Intepting we have
~(<~<~+<+<M'&)==<W.
Now from the precedmgvalues
But the rigbt-hand 9tde of the equation dénotes the rec!pr<cnlof the square of a central radins whose d!to<~ion-cos!aesaredx <&
~'<&'<&'·
The geometric meaning therefore of the integral wo hâvefound!a ~J?=*constant.*
348. The constant~D ~<Mtlie «MMvaluefor aU ~M~MCatoAM~~OM throughan umbt7ic. For at tho nmMIIo the p ia
of com'Becommon to aU, being =Y and since the central
section parasol to thé tangent plane at the umbUtcis a cirde,the diamoter parallel to thé tangent line to the geodestc îs
constant being always equat to thé mean ax!s b. Hence fora géodésiepasaingthroughan umbilie,we havejpD~ao.
Dr. Hert proveathesametheoremas6)Uov<:Oonstdermyplanesectionof an ellipsoid,têt betheperpendicnlarfromthe centreof thesectionon the tangentUne,d thediameterof thesectionparallelto thattMtgent) théangletheplaneofthe sectionmakeswiththe tangentplaneat anypoint. Thenalongthe Metion Mconstant,and it MevidentthatpD iaina nxedratioto w<!sin<.Henee alongtheaeetionpDvariesaa siniand wtUbe a taaxhnmnwherethe ptaaemoetathe Mr&eeper-pendioularly.Bata géodésieoset~tesa aenesof no'nMdseetMMtthere.fore, for sucha linepD la constant,its differontialalwaysvanishing.ÛtH<ttK~«and DublinJ!<o(A<tHa(<e<~J~MnM<,VoLiv., p. 84.
CURVESTKACKDONSURFACES. 285
Let now any point on a quadric be joined by geodesicstotwo ttmhttics,since we have just proved that pD Mthe same
for both geodesics,and mace at the point of meeting the p is
the same for both, the D for that point must atso have thommo value for both; that is to aay, the dtameters are equalwhich are drawn paraM to the tangente to thé geodesicsat
their point of meeting. Bnt two equal diameters of a con!c
make equal angles with !ts axes; and we know that the axesof the central sectionof a quadrio parallel to the tangent planeat any point are parallel to tho directions of the lines of cm'-
vature at that point. Hence, the ~eod!M<M~MKM!~any pointon a ~K<K~Mto two M~M~CNmake€~<M?angles <M'<&the lime
of curvature<XMM~that p0!t!<It 6)!lowsthat the geodesicajoining any point to the two
oppositeumbilics,which lie on thé same diameter, are con-
tinua~ons of each other; sinco the vertically oppositeanglesare equalwhich thèse geodesicsmake with either lino of cur-
vature through the point.It followsa!so (aeeArt. 848) that the M<Mor <?~MKMis
CMM<(Ht<of the ~coJ<MMdistances of all the ~otK~ on the aame
lino of curvature from <~M«mAf~tM. The sum is constant
when the two mnMUcschosen are mtcrior with respect to thé
Uneof eurvature; the differencewhen for one of these umbilles
we substttate that diametrically opposite so that one of thé
nmMUcsis interior, the other exterior to tho line of curvatare.If A, A' be two oppositeumbilics,and B another umbilic,
since the sum PA+PB is conatant and a!ao the dinerence
PA'-PB; it Mows that PA+PA' Is constant; that is to
say, all the ~M~eatOtM'%<cAco~)tec<<MOo~pM~e«MStKcsare
of egM~ &~<& In fact, it is evident that two indefinitelynear
geodesicaconnecting the same two points on any aur&cemust
be equalto eachother.
849. Me constantpD has the same c<!?«efor «H ~eo<~MM«)~t'< touckth eame line of curvature.
Thh theorem and itt conMquencM devetcped in the followingarticles
are due to Mr. Michae! Roberte, I.iouvi))e, Vol. X! p. 1.
CUBVE8TBACED ON BOBPAC~.286
It was proved (Art. 174) that pD hae a constant value all
along a line of coi'vatuM; but at the pointa where either
geodesic touches the Uneof curvatureboth p and D have thesame vaine for the géodésieand the line of curvature.
Hemce then a eystem of Unes of curvature bas properdescompletely analogousto those of a system of confocal oom~in a plane; the umbilicsaMwenng to the foci. For example,<!<?~ax~MtCtangent8<~(MC!!<0one~OM any point on anothermakee~tM~o~&e <c&&the tangentat <%<!<point. Dr. Gravee'etheoremfor plane conics holds a!sofor lines of curvature, v!z.that the exceas of the snm of two tangents to a line of enr-
vature over thé interceptedarc Mconstant,whilethe intersectionmovea along another line of curvature of the same 6pec!es(aee<~M«M)p. 297).
MO. The equationpD = constanthas beenwritten inanotherconvenientjbrm.* Let d, a" be the primary sem!-axesof two
confucalsurfaces throngh any point on the curve, tmd let t be
the angle which the tangent to the géodésiemakeswith one ofthe principal tangents. Then since<«", a*-o"*(Art. 172)arethé sem!-axesof the central sectionparallel tothe tangent plane,any other Bem!-d!ameterof that sectionis given by the equation
861. ?Xe <o<!<Mof ~e tM&MM<MMto~ <Mo~eo<~<Mctangentetoa lineof curvature,M~tbAca<at aMyXM,Ma <pJ5<M-CMtM.
Th!s is provedas the correspondingtheoremforplane cornes.
Ifa', a" belongto the point of intersection,wehâve
a" coa't+ot"*Nn*~==comtaBt,a" a!n'<+a"' cos't's constant,henoe a" +a'" = constaDt
ByI.MuTtt)e,Yo!.tx.,p.401.
287CBRVE8Ttt~CED ON SUBFACES.
and therefore(Art. 169)the distanceof the point of intersection
&omthe centre of the quadricis constant. The locus of inter-section is thereforethé intersection of the given qoedric witha concentricsphere. The demonstrationholds if the geodesicsare tangents to difRerentMnes of eurvature; and, as a par-ttcnla!'case, the locuaof thé foot of the geodesicperpendicularfrom an umbilic on tho tangent to a line of curvature is a
ephero-conic.
852. To~M< ?<?«<<~ <H<M'aee<MM!<~~eodM: <<M~<K<a? a line of curvature M~M~CM<at a ~tOMtangle (Be~O)HonvIUe,XtV.p. 247).
The tangents from any point whose a', «" are given, toa given Une of curvatnre are determined by the equationa" cos't+a"* Bin't=~3; and since they make equal angles witheither of the principal radii throagh that point, t thé anglethey makewithone of these radii is half the angle they makewitheachother. We hâve therefore
whenco it appem that the locusreqairod Mthe intersection
of the quadriowith a snr&ce of the fourth degree.*
853. It was proved (Art. 186)that two confbcals Cttnbedrawn to touch a given line; that if the axes of the three
surfacespMsmg through any point on the Ime be <t,o', a"
and the angle the line makes with the three normals at the
Mr.MichaelRobe~ hMproved(Liouville,Vol.xv.,p. 291)bythemethodof Art.19~,that the projectionof this curveon the planeofcireularaecdoMis the toctaofthe interMethmof tangenteat a conatant
angleta theconicintowhichthelineof curvatureis projected.
CUBVES TRACED ON SURFACES.288
point be a, <y;then the ax!9-m~orof the touched con&calie determined by the quadraëc
Let us suppose now that the given line Ma tangent to thé
quadric whoseaxis is <~we hâve then coaa~0, since the lineis of courseat right angles to the normal to thé first surface;and we have coa~3=9!ny,amcethé tangent plane to the Mr-&ce a containsboth the line and tbe other two nonnalB~ The
angle </is what we have called i in the articles immediatelypreceding. The axis then of the secondconfocaltouchedhythe given Hne is determinedby the equation
If then we write the equation of a geodcsic (Art. 35!)0" cos*~+<t"*sm't==a*,we see from this article that that équa-tion expressesthat all the <aKyeK<&KMalongthe sameyax~tc<OMcAtheCOM/OCC~M~ceM~Me~M'Ma~axis M&
The geodesicitse!f will touchthé lino of curvature in whiehthis confocalintersects the original sar&ce; for the tangentto thé geodeMCat the point where the geodesicmeets the
confocal ts, as we have just proved, also the tangent to tho
confbcal at that point.. The geodesictherefore and the inter-section of the confocaland the given surfacehave a common
tangent.The oBco!atmgplanes of the geodesieare plainly tangent
planes to the same confocal; smoethey are the planes of twoconsecutive tangent Unes to that confocal.
The value of pD for a geodesic passing through an
umbilic is ao (Art. 848); and the corresponding equationia therefore <t''eoB''t+a"*sm't=a'-t'. Now the confbcal,whose primary axis is '/(«'-5*), reducesto the mnMHcarfocalconic. Hence,as a particularcaseof the theoremsj~t proved,
The theorems of this article are taken &om M. Chasle<'<Memoir,
L!oMTi)t<Vo).XT.,p.
CURVES TRACED ON 8URFACE8. 289
all &M)~eK<&nM a ~eo<~MM<oXMAy<M<M~Mt~A an umbilic,tKterMetthe «mJMHMr~oa~ce)!&
Conversely, if from any point 0 on that focal con!c recti-
Unear tangents be drawn to a quadric and those tangents
prodcced geodet!caUy on tho sur&ce, the lines so prodncedwill pass through the opposite umbilie; thé whole lengtbsfrom 0 to thé umbilic being equal.
854. From tbe &ct (proved p. 144) that tangent planesdrawn through any line to the two con&cals whtch touch it
are at right angles to each other, we might have inferred
direetty, precisely aa at Art. 279, that tangent lines to a
geodesic touch a confocal. For thé plane of two consécutive
tangents to a geodesic being normal to thé surface is tangentto thé confocal touched by~tho 6rst tangent. The second
tangent to thé geodesic therefbre touches thé samo confocal;
as, in like manner, do all the succeedmg tangents. Havingthus established the theorem of the last article, we could, byreveraing thé steps of the proof, obtain an indepcndent de-
monstration of the theorem pD =constant.
8S5. The t&M&paMeeMMMMMrtMto a g~o~c along a
~e<x~M<o&M ita cuspidal edge on another ~aJnc, <e~cA M
same for all ~KodiMMStouching the same line of curvature.
For any point on the cuspidal edge is thé intersection of
three consecutive tangent planes to thé gtven quadric, and
the three pointa of contact, by hypotbesis determine an oscu-
lating plane of a géodésie which (Art. 8M) touches a 6.ted
confocal. The point on the cuspidal edge !s the pole of this
plane with respect to the given quadrio; but thé pole with
respect to one quadrio of a tangent plane to another lies ona third fixed quadric.
8S6. M. Chasies haa given the Mlowing generat!za6on ofMr. Boherts's theorem, Art. 848. If a <&Mo~~M<MM<?at (Mû
jS!M<<J)0m& OMCM gtMK~tb <<M! by a ~eK<~MOeM~along a coM~xxt!B (so that the thread of course lies in geo-desics where it Is in contact with the quadnca and in rightUnes in the spaoe between them), <~e)t tho ~enct! toill ~-aoe
CURVES TRACE!) ON SURFACES.290
a line of CMyMtttMon the gtMM~fMA. For thé two geodesicson thé surface J3, wMeh meet in tbe tocua point J~ evidentlymake equal angles with the tocns of P; but these geodesicshave as tangents the rectilinear parts of the thread which
both touch the same confocal; therefore (Art. S53) thé pD is
the same for both geodestcs, and hence the Hne Msectmg the
angle between them Is a line of curvatnre.
A particular caae of this theorem is that the focal ellipseof a quadric can be deMnbed by means of a thread fastened
to two fixed points on opposite branches of the focal hyperbola.
357. ja?~«:(~~M<t<M. The method uaed (Arts. SS1,352)in which the position of a point on the elUpsoid is denned bythe primary axes of the two hyperboloide intersecting in that
point, M called the method of EUiptic Co-ordinates (see p. 162
and JB~Xef Plane C'Mf))M,p. 276). It being more convenient
to work with unaccented lettera, 1 follow M. Houvule* in
denot!ng the quantities which we have hitherto called a', a"
hy the !ettem ~t, f; and in tMs notation the equation of the
lines of curvature of one system would be of the form
~==constant, and those of the other t"= constant. The equationof a geodeaie (Art. 350) would be written cos'<+ Mn't==~ iand when the geodestc passes through an umbilic, we have
=~&'=A*. It wUl be ramembered (Art. 166) that Kea
between the umits h and k, and f between the Kmits k and 0.
Throwing the equation of a géodésie into the form
we aee that it is satisfied (whatever be /t') by the values
/t*=)' tan*t=-l. Whence it follows that the sMae pair of
imaginary tangents, drawn from an umbilic, touch aU the Unes
of cnrvatut'e,! a further analogy to the foci of plane conice.
358. To &BpMM M eN~<M co-<M'<?MM:<e<the element of the
arc o~ any eurve on the <tt)~tce. Let us consider first the
1 cannot, ho~eYei, bring myeeMto imitate him in calling the M!s
of the eH!pM)M and Ms denoting the q~ntittea <t' b', «* e*(whichwe ceU & ~) by the lettera b', < Mem<liiM~ to confuse.
-)- Roberts, Liouville,Vol. xv., p. 289.
H2
elementof any line of curvatnrc, = constant. Let that l!nc bemet by the twoconsecutivehyperboloide,whoseaxes are v and
f+A; then, since it cut8 them perpendicutarty,the interceptbetween them is equal to the difference between the central
perpendicalamon the tangent planes to the two hyperbolotds.But (Art. 190) (p"+<)'-p"'=(f+Jf)' or~Now we bave proved that <=<?<r, the element of the arcwe are seeking, and
Now if through the extremities of the element of the arc <&
of any curve, we draw lines of curvature of both systems, we
form an elementary rectangle of which <i'o, J</ are the sides
tmd da the diagonal. Henee
359. In like manner we can express the area of any portionof the surface bounded by four lines of earYatare; two lines
~) and two f,, f,. For the élément of the area is
The area of the s(u'&ce of the et)ip<otdwas thus NKt expressed byLegendre, ït~tt~ &< J%Me<tM«Elliptiquas, Vot. ï., p. 362.
CHRVES TRACED ON SURFACES.292
orthogonal trajectory of a curve whose d!6erenttat equation M
~H~+jKHf. For tho orthogonal trajectory to JR&r+C~' M
plainly 'B sinee J</ are a system of rectangalar
co-ordmates. Bat 3&~ + Ndv can be thrown without difficultyinto the &rm JMtr+~<r' by the equations of thé last article.
The equation of the orthogonal trajectory is thus found to be
860. The fnt integral of a geodesic co9't+~ Nn'<=!can be thrown into a form in which the variables are separatedand the second integral c&nbe obtained. That equation gives
861. To ~M an &~p!'MMMtfor the length of OKyportion ofa geodesic. The element of the géodésie M the hypoténuse of
a right-angled triangle of which d~, dd are the sides and whose
baM angle is i. Henee we hâve <&=!8mt<Hf'ico:t'<&!r;and
CUNVE8fRACBD.ONSURFACES. 293
It is to be noted that when wa give to the radical in the last
article thé aign 4- we must give that m this article the signThis appears by forming (Art. 359) the dïBerential equation ofthe orthogonal trajectory to a géodésie through an umMUc, an
cquation which must be equivalent to <~='0 (Art. 842).
S62. In place of denoting the position of any point on an
ellipsoid by thé elliptic co-ordinates ~t, f, we might use geodesic
polar co-ordinates and denote a point by p its geodeaic distancefrom an umbilic, and by m thé angle which the radius vcctor
makes with the tine~joining the umbilies. Now the équation(Art. MO) of a géodésie passing through an umbilic gives tbe
aum of two intégrais equal to a constant. This constant can-
not be a fonction of p since it remains the same as wego alongthe same géodésie it must therefore be a function of m onlyand if we pass from any point to an indefinitely near one, M<~
on the same géodésie radius vector, we shall have
We shall determine the form of the function by calcnt&tmgits value for a point indefinitely near the umbilic, for whieh
~t=!y=t~. The left-hand side of tho equaëon then becomes
Now since the angle externat to the vertica! angle of thé
triangle formed by the line joining any point to two amMUcs,Mbisected by the direction of the line of curvature, that externat
angle is double the angle i in the formula coa~t+t'* sm*~ &In the limit when thé vertex of the triangle approaches the
294 CCBYESMACEDON8UKFACE8.
863. If P, Q be two consécutive points on a cnrve, and if
PP be drawn perpendieular to the geodesie radius vector OQ,it M evident that JP~'=JP~+jP'<y. Now since (Art. 342)
OP=OJP', we hâve F'a~, while JPP' being the element
of an M'o of a geodeaio circle, for which p is constant (or
dp = 0), must be of the form J~M. Hence the element of the
arc of a curve on any surface can be expreMed by a formula
<&'==<+JP'<&e'. We propose now to examine the form of
the fonction P for the case of radu vectores drawn throughan umbilic of an ellipsoid. Let us consider the line of our-
vatnre /*==/ We have then (Art. 36!)
CUBVES MACED ON SURFACES. 2M
In this investigation it is not noceseary to assume the result
of the last article. If we substttute for the right-hand aide of
the equation in the last article an undetermined function of <a,It is proved in like manner that JP=y~(«'). We determine
then tho form of the funct!on by remembering that in the neigh-bourhood of the umbilic thé surface approaches to the &nN
of a sphere. Now on a sphère the formula of rectification
is <&'==<+8m'p~ Renée JP=a!np. But in the sphere
y'sinpsmM. The fanction therefore which multiples y is
1
ainM*
364. Consider now the triangle formed by joining any
point P to the two umbilics 0, 0'. Then for the arc OP
we have the function Pc=- and for the Mo O'JF~connecting81nœ
P with the other umbilie, we bave the funetion = i
and J°: JP* stBM smM', an equation analogous to that which
expresses that the aines of the sides of a spherical triangleare proportional to the sines of the opposite angles; since P
and .P* in the rectincatton of arcs on the eU!psoidanswcr to
s!np, sin~' on the sphère.
365. Aga!n, if P be any point on a Imo of curvature we
know (Art. 348) dpi~'=0, where p and p' are the distances
from the two umbilics. Now if <? be thé augle which the
radius vector OP makes with the tangent, the perpendicularelement B~M is evidently dp tan~. But tho radius vector O'jP
makes atso tho angle with the tangent. Hence, we have
CONTESTRACEEON8CBFACES.296
whence tan~Mtam~M*Mcomatamtwhen the Mtmof sides ûf the
triangle is given; and tan~m is to tan~M' in a given ratiowhen the differeneeof aides of the triangle is given. Thuatben the distance betweentwo umbilics being taken ae thebase of a triangle, when either the product or the ratio ofthe tangents of the halveaof the base angles is given; thelocnBof vertex is a !ine of cnrvatme.*
From this theorem follow many corollaries: for instance,If a geodeaicthrough an umbilic 0 meet a lino of carvatme
in points P, f, then (accordingto thé specieftof the lineof curvature) either the product or the ratio of taB~JPO'O,tan~JP'0'0 is conBtant." Agarn, "!f the geodesicsjoining tothe umbilics any point P on a Une of curvature meet théeurve again in P', P", the tocns of the intersection of the
transverse geodesicsO'JP',OP" will be a line of curvature ofthe same species."
866. Mr. Roherh's expressionfor the element of an an:
perpendicular to an umbilical géodésiebas been extended M
follows by Dr. Hart: Let 0~ OT*be two consecutivegeo-deaiestouchingthe line
ofcurvatureformedbythe intersectionof the
surfacewitha confocal
B, <~Mthe angle at
which they mtersect;then the tangeht at
any point T of either
géodésie touchesB ma point P (Art. 353)¡and if TT' be taken
conjugate to 2T~ the
tangent plane at T*
passes through TP
This theorem, as well as thoMon which its proof dépende,(Art. 362,
&o.) due to Mr. M. Roberts, to whom this departmeat of Oeomettyowes so much tLiooviUe,Voh. xni., p. and xv., p. 2T&).
90~<Kf<CURVES TRACED ON SURFACES.
(Art. 287) and thé tangent !me to the géodésieat f touchesthé confbcaiB in the same point P. Wo want now to expressin the form JM<ethe perpendicular distance from y to TP.Let the tangents at consécutivepoints, one on each géodésie,intersect in P* and make with each other an angle o~ Letnormals at the points 2~'meet the tangents JP?~ FT' atthe pointe2~, T, then sincethe dtaërence between ~) 2~'is infinitely smaUof the third order, PT,~ and F'T~' are
equal to the aame degree of approximation. But PT.,are proportional to D and 2)' thé diameters of the snr&ceB drawn parallel to the two successivetangents to thé geo-désie. Hence D<~=J)'< This quantity therefore romainsinvariableas we proceed along the géodésie; but at the point0, <~=<?<a if therefore Do be the diameter of B parallel tothe tangent at 0 to the géodésie, 2M~=D~m; and there-
fore thé distance we want to express JP?!~==~<~M, where
t (=PT) is the length of thé tangent from T to the confocalB;
or <!s a mean betweenthe segments of a chord of B drawn
through JTparallel to the tangent at 0. When the géodésiepassesthrough an umbilic, the surface B redncesto the plane
of the umbilics,and t becomesthe line drawn through T
to meet the planeof the ombilicsparallel to the tangent at 0;iwhichisMr. Roberts's expression.
Rence, if a ~e<x?(Mt'c~c~OHcM'CMMMWSea lineof curvature,and if all Meangla but one moMon &MMof curvature,thiaa&o<Ct!!MOMona lineof CM'MfM'<and the~MMtMetefûjfthepolygon<o<?Jbe constant when~e ~Mteso/' curuature are of the same
opecMt.The proof Is identical with that given for the corre-
spondingproperty of plane conics (G~MW,Art. 358).
367. If a géodésiejoining any umbilicto that diametricallyopposite,and making an angle «*with the plane of the um-
bilics, be continoed so as to return to thé first umbilic, itwiU not, as in the case of thé sphère, return on its former
path, bnt on its return will make with the plane of the mn-
billes an angle dinërent from M. In order to prove this we
CUBTB8 TRACED ON SURFACES.298
ahtt!! investigàte an expression for <?,the angle made withthe plane of the amMMcsby the oseulating planeat any pointof that geodemc.
It is convenient to prefix the following lemma. lu a
sphencattnamgleletoneddeamd the ad-
jacent angle remain imite while the basedumn!ahesindefinitely,it is required to fmdthe limit of the ratio of thé base to thé
déférence of the base angles measnred in
the same direction. Thé formulaof spherieal
trigonometry cos~(~-t-J9)=8m~C'L gives us in theCORC
limit <?<9=cosa~ But evidently sioa~MB~ Hence
<~mn3 ttma
Now we know (Art. 8S3) that the tangent line at any pointof a geodesic passing through an umbilic, if prodaced goea to
meet the plane of the umbilica in a point on thé focal hyper-
bola and the oseulating plane of the géodésie at that pointwill be the plane joining the point to the corresponding tangentof the focal hyperbola. We know also (Art. 194) that the
cone circamacribing an eUIpMidand whose vertex is any pointon the focal hyperbola is a right cone.
Let now ~'P' be an element of an umbilical géodésie pro-duced to meet the focal
hyperbola in jB. Let
JP'J°" be the consécutive
element meeting the focal
hyperbola in H'; then
if J5~, jETA'be two con-
sécutive tangents to the
focal hyperbola; PHh,P'Eh' will be two consecutive oscntatmg planes. Imaginenow a sphere round J?', and conaider the spherical triangle
formed by radii to the points h, h', P. Then if <~ be the
angle ~C' the angle of contact of thé focal hyperbola;thé angle between the oseulating plane and JU?'&'the plane
of the umbilics, while AN'P* is the ecmi-aNgte of the cone;
CURVES TBACBD ON SUBfACES. 299
themthe a{Aen~ triangle M that comidered la ovr lemma,
and we have =smp tana
In order to integrate this eqaat!on we must expressJ~ interms of a; and thia we may regard aa a problem in planegeomehy, for a MMf the angle mdaded betweenthe taageats&om J5rto the principal section in the plane of the umbûtca,while <~ ie the angle of contact of the focal hyperbola&t thesamo point Now if a', &' e", &"be the axes of an ellipseand hyperbola paasingthrough H, confocalto an ellipsewhos6axea are a, &; and if 2ctbe the angle ineluded between thé
tangents from H to the latter ellipse, we have (ace CbMM~,
Ex. 10,p. 192) tan'ot=- Differentiating, regarding o"
as constant (sincewe proceed to a consécutivepoint along the
same confocal hyperbola),we bave <&['="-tanct -)s' But
if ~)' be the central perpendioalam on the tangents at H
to the ellipse and hyperbola, we have &'<&<'=~(~(Art. 858).Now dp Is the element of the arc of the focalhyperbola, and
if p be the radius of curvatmre at thé same point, t~p=p~a'a"* tr <<~
Bato=. Hence<&t=:–tana~-7'-or<&t=tana–7M-.
In the case under consideration the axes of thé touched
eHtpaeare (!, c; while the squares of the axes of the confocal
hyperbola are a* & &* c*. Hence we have thé équation
Intcgratmg this, and taking one limit of thé integral at
the umbilic where we hâve = at, and a = we have
300 CUBVES TtiACED ON SURFACES.
If then Jt ho the value of this integHd; we have
tan~~&tan~, where i!:=~.
Now this integral obviously does not change sign between
the limite i~, that is to aay, in passing from one umbilic
to the other. If then M' be the value of 9 for the umbilic
opposite to that from which we set out; at this limit I haa
a value different from zero, and k a value different from unity;and we have tan~M*==A*tan~m m'M therefore always dIBerent
from M. And in like manner the géodésie returna to the original
~unbiltc, making an angle M" sach that t!m~w"==A*tan~and so ït will paas and repass for ever making a sénés of
angles thé tangents of whose halves are in continued pro-
portion.*
868. If we conBtder edges belonging to the Bame tangent
cône, whose vertex M any point H on the focal hyperbola, a
(and therefore k) is constant; and the equation tan = &tançât
givead8 dm
Nowsince the la plane of thegives
= Now since thé oseulating plane of thé
geodesic is normal to the surface, and therefore ako normal
to the tangent cône, it passes through thé axis of that cône.
If then we eut the cone by a plane perpendieular to thé axis,
the section M evidently a cirde whose radius !s -~L and thesin 0
element of thé arc ia or 4–. Now this element, beingsmC' Stna)
the distance, at their point of contact, of two consécutive aides
of thé cireornscribing cone, is what we have called (Art. 363)
jMm, and we hâve thus from the investigation of the tast
article an independent proof of the value found for P (Art. 868).
369. Lines of level. The ineqnalitles of level of a countrycan be represented on a map by a sénés of curves markingthe points which are on the same level. If a acnés of such
curves be drawn, corresponding to equi-diSerent heights, thé
The thect-MMof thit articleare Dr. Hart's, Chttth~e and J0<(&~JM!.Mtmn<«!tt~JeMf))~Vol. iv., p. 8!) but in the modeof proof 1 hâvefoUowedMr. WiNttmRobert*,HouviBe1867,p. 213.
CUKVE8 TRACED ON 8UKFACM. 801
placeswhere the cnrvea lie closeat together evidently indicatetheplaceswhere the level of the country changéemost rapidly.
GenertJty,the curves of level of any surfaceare the sectionsofthat surfacehy a seriesof horizontal planes,whieh we maysupposeall parallel to the plane of a~. The equations of thehorizontalprojectionsof such a aeneeare got by putting <=<'(!in thé equationof thesurface; and a dinerentiatequationcommontoaU theseprojectionsis got by putting <&=0in the differential
equationof the surface,whenwe hâve
We can make this &function of x and y only, by eliminatingthe e which may enter into the d!Serent!at coefficients, by the
help of the cquatton of the surface.
ZMMSo~~ea(M(a&)pe. TheHneofgreatestalopethroughttny
point is the line which ente all the lines of level perpendicularly;and the differential equation of its pMgec~ontherefore tB
The line of greatest slope is oftendefinedas that, thé tan-
gent at every point of which makes the greatest angle with
the horizon. Now it is evident that the Une in any tangent
plane which makea thé greatest angle with the horizon isthat which is perpendicular to the horizontal trace of that
plane. And we get the same equation as before by exptesMngthat the projectionof the element of thé curve(whosedirection-cosineaare proportionalto <& dy) is perpendicularto the tracewhoseequationis
It la évident that the differential equationef the carw, wMch hthf~ya pefpeBAieattîto the intemetion of the tangent piffne, jwhoMditeetion~omnesare as JE,JM,~V] by &&Md plane whoMdirect!on-<MinM
11-1.-
CURVESTRACEEONSURFACES.802
Et. To&tdthe line ofgKatMtetopeenthe qnadtieji~tB~C~~D.The d!9i'rentialequationis ~J! whieh iate~ated, gives
(~) ° (~!wherethe constanthas been determinedby the condition
that tbe !me dtftti pass through the point <='< y y'' The Uneof
greatest dope is the intersectionof the quadricby the cylinderwtose
equationhMjust beea written,and willhe a curve of doubleeat~tute
except when lies !a one of thé principalplanes when the equationjust foundredncetto ~=0N'yO.
370. We ehalt conclude this chapter by giving an accoant
of Gaum~ theory of the curvature of Mr~cea.* In plane curves
we measure the eurvature of an arc of given length by the
angle between the tangents, or between the normab, at its
extremMea in other words, if we take a c!rcle whose radmais unit y, and draw radii parallel tu the normals at the cx-
tremities of the are, the ratio of the intercepted arc of the
circle to the arc of the curve affords a measure of the cor-
vature of the arc. In like manner if we hâve a portion ofa surface hounded by any closed curve, and if we draw radii
of a unit Bphere parallel to the normals at every point of the
bounding cnrve, the area of the corresponding portion of thé
aphere is called hy Gauss thé total curvature of the portionof the surface under consideration. And if at any point of
a surface we divide the total curvature of the :nper6cial element
adjacent to the point by the area of the clément itsel~ the
quotient is called thé m<N<MMof curvature for that point.
371. We proceed to express the measure of curvature bya formula. Then since the tangent plane at any point on the
surface, and at the corresponding point on the unit aphereare by hypothesis parallel; the areas of anyelementary portionson each are proportional to their projections on any of tho
co-ordmate planes. Let us consider then their projections on
the plane of :ey, and let us suppose tbe equation of the surface
to be given !n the form et=~(.c,y)
The reader willSnd bis paper repnnted ia the appendix to LioaviUe'oedition of Monge.
CURVE8TBACEP OHSURPACEa. 803
If then te, y, jebe the co-ordinatee of any point on the surface,
J~ y, thoae of the corresponding point on the unit aphere,
a!+<& a;+&B, ~'+< ~+& &c., the co-ordinates of two
adjacent points on each then the areas of the two elemcnttuy
triangles formed by the pointa conmdered, are evidently in the
ratio
Now .1, Z being thé projecdom on thé axes of a unit
line parallel to the normal are proportional to thé cosines of
the angles which the normal makes with the axes. We hâve
there6)ro
But from the equation of (Art. 281, p. 222) it appears that
the value just found for the <m€<MM~eof CM~M~MMM~n,,
where
<t<tJ <t)'e~e t«'o ~ftHC~M?)'<)!<?o/'cMrm!~<tv<t<~<!jMM~.
CUKVE8TRACEDON6CBFACE8.804
372. It is easy to verifygcometrieallythe value thus found.For consider the elementaryrectangle whose sides are in the
directions of the principal tangents. Let the lengths of thésidesbe X',and conMquentiyits area New the normalaat the extremitiesof intersect, and if they make with each
other an angle we have ~=~ where is the corresponding
radius of carvatnre.- But the corresponding normals of thé
aphete make with each other, by hypotheiiis,the same angle;iand their length is unity. If therefore be the length of
the element on the apherecorrespondingto X, wehâve p==~.
In like manner we have-i~=~
ande=
wMch wasAA. ~~t
to he proved.878. Gausa has proved that if a mrface eapposedto be
flexiblebut not extensiblebe deformedin any way (that is to
say, if the shape of the surface be changed, yet so that thedistance between any two points meastiredalong the surfaceremains the aame) then the measure of eurvature at everypoint remains unaltered. We have had an example of aucha change in the case of a developablesurfacewhich ïs moh a
deformationof a plane (Art. 287). And thé measure of cnr-vature vaniahea for the developableas well as for the plane,one ofthe principalradii being infinite (Art. 884). To establishthe theorem in gênerai, let us supposethat any point on thesurface instead of being given by three co-ordinatesconnected
by the equation of the surface is given by two independentco-ordinates. Let
what we want to provo is that the measure of curvature, orthat the productof the principalradH,ia a fonctionof JP,J~ G.
CURVESTRACEDON SCBfACM. ,ao5
x
In &ct, let a:yz' denote the point of the deformed surface
correBpondtogto any point .B~ of tho given Bur&co. Then
a: y', < are given fnnctionaof a', y, < and can thereforeataobe expremed in terms of u and e. And the element of anyare of the deformedsurfacecanbe expt6Baedin the form
But the condition that thé length of the arc shall be un-altered by transformation,manifestly requirea F=JE', JF'=~6'==<?'. Any function therefore of JE, J~ <? is nn<eredbysuch a deformationM we are conaidering.
Now it will be remembered(aeep. 20!!)that thé principalradti are given by a quadratic, in which thé cocf5c!ectofis (L'+JM''+jV')'; and the absolute tenn is
We shall aeparatdy express each of these quantities in
termacfIi',F,<?.
374. Now if we subatitute in the equation of the mrface
Z<ib!+3~+jM&=0, thé values of <&c,a~, <&given in the
last article, tmd remember that smce u and v are independentvariables,the coefficientsof <&tand dv must vanish separatety,we have
(SeeZesMMMon ~%f~ef~l~e~Mt,Art. 21).
875. Let us now examine the result of making in thoabsolute term, given Att. 873, the Mumesabstttution, ~z.
We use Roman letters in order that the a, &,o of p. 203 may notbe con&tmded with <<,&,<*used in a differentMMe in this article.
CURVES TRACED ON SURFACES.306
Z=\(&c'–&'<;), &o. Now an équation whieh we had oecaa!on
to use in the theory of comca(eeeCMMM,Ex. 5, p. 2M) enables
us to write this result in a more simple form. Let ua writo
down the equationof a conie
In &ct, either side of this equation, equated to nothing, ex-
presses the condMonthat the line joining the pointa abc,a'&'c'ahouldtouchthe corne. The equation however may bo venned
by actual multiplication. What we want to calculate then is
~'(P'F*)wheM
and making these anbatitntions on the left-hand aide of the
precedmg cquation, substitute for <&,dy, &, from Art. 373,we get, by equatingthe coefficientsof (?M*,~M~ and de,
CURVES TBACED ON SUBFACES. 307
xs
Now if theae prodnctsbe expanded accordmg to the ordinaryrule for tm)Mpttcat!onof determinants, they give thé d!Setence
betweenthe two determinants-
376. Now it is easy to show that the terme in these deter-
minants are 6mct!ona of J~ <? and their dt6erent!atB. Re-
ferring to the definitions of < &,c, a, a', a", &c. (Arts. 873,875)itMobvioasthat
1 owe to Mr. WîUiMMon the remark that the application of this
rule etMMta the KMit in a form whieh manifesta the truth of GfHtM'a
theorem.
CUBVESTRACEDOSNURFACEa.808
It wIHbe aeenthat these equationsexpress in terms of E, J~ (?
every term in thé preceding determinants exeept the leadingonein each. To oxt)<'<'as.hese,diSerentiate, with regard to ?,the equationla~tTviliten,and wehave
tNow hecaaso
da= ) &c<)the quantities withinthé bracketsow
dv `~ duy t q quantltlesW1 ID e nwAets
in the last two equations are equal. And smee the leadingterm in each determinant !Bmultiplied by the Bameiactor, in
subtractingthe determinants we are only concernedwith thedjNërencoof thèse terms, and the qnantity within the hraekets
disappearsfrom the result. This reeult is multiplied bythe differenceof the determinants
<md
We get the meMnïe of carv&toNby dividing thé quantitynow&)me~ by (Z'+Jf+~)' whosevalue is given(Art. 874)when the commonfactor disappears and the reealt ia oh-
CURVESTBACEbONSURFACES. 809
TNuatya functionof E, J~ <? and their diSerentîa! Gausa'atheomm !a therefore proved.
We add the actual expansion of the determinants, thoughnot necemaryto the proof. Writing the measureof curvaturo
j5, wehave
377. We may consider two systems of curves traced on
the surface, for one of which u is constant, and for the other v-1M that any point on thé surface !s the intersection of a cnrve
of each ayatem. The expression then <&=~M'~ 2F<~<~c+(Mp*
shows that V(~) du is the element of the curve, passing thronghthe point, for which v is constant; and V(~)~ is the element
of the curve for which u is constant. If these two curves
intersect at an angle M, then since <&is the diagonal of a
parallelogram of which ~(jE)~M, ~(<3)~c are the sides, we.E*
have cosa='T<
while the area of the parailelogram being
<~<&/sinm= \~(JF<? J!") dudv. If the curves of the system u
eut at right attgleBthose of the system c, we mmt have jF= 0.
A particular case of these ibrmuhe is when we use géodésie
polar co-ordinates in which case we saw that we always have
MM. Bertrand, M~oet, and Pa:aeax (eee Lioaifi!!e,VoL xin., p. 80;i
Appendh to Mo])~, p. 683) have eatablished GauM'atheorem by ealett.
lating the perlmeter and area of a geodeHe eMe on any «tt&ce, whoM
mdtOB,mppoKd to be very amall, is <. They Cad for the perimeterM't~
SM =~s;t and for the area M* And of course the supposition8jt~ t~~t~t
that thMe are unaltered by déformation impliea that JHf is constant.
CCNVE8TRACEDONSURFACES.310
an expressionof the form <&'=<+.PW. Now if in the
formuléeof the last artMe we put ~-0, ~~conatamt, it
becomes
an equation which mtist be satisfied by the function P on any
surface, if JMt! expresses the element of the arc of a géodésiecircle. Mr. Roberts verifies (Oambridge and Dublin JHat<&e-
NM<M<t?JoM~o~ VoL in., p. 161) that this equation is satisfied
by the fonction on a quadric.sin.
378. Grauaaappliesthèse ibmmiœto 6nd the total corvatnre,in his senseof thé word, of a geodea!ctriangle on any aar&tce.Thé élément of the area being JMm~p,and the meMNKof
curvatarebeiBg -D-jr;
the total curvatare is found by
J'P P.:J.7 ln J!- ,thtwice integratmg f~M. Integrat!ng &'st with reapect
to we get (<7- ,-)af~. Now if the radu are meaBured
6'om one vertex of the given triangle, the integral is p!amlyto van!ahfor p=0; and it Mplam also that for p ==0we must
have=1
for as p tends to vantsh, the lengthof an élément
perpendicnlarto the l'adms tends to becomep<&t. Hence the
&mtintegral !a<~1 ,-)
This may be written in a more convementform as &)Uowa
Let 9 be the angle which any rad!na vector makea with the
element of a geodesic ab. Now
sinceoa'=m<a,M'=(F-t<U')<J'<a;and
if e&=<M',wehâve &'c= <&M<M,and/yp
the angle y<M=-y-<&?. But bac N
evidentlythe diminutionof theangle
CUKVK8 TNACED ON SURFACES. 311
in passing to a consécutive point; hence <7~= -j- do. The
integral just found is there&re J<a+~, which integrated a
second time is <e+ C' 6", where m is thé angle between the
two extreme rad!t vectores which wo consider, and 0', 6" arethe corresponding values of If we call J?, C the internai
angles of the triangle formed by the two extreme radu and
by thé hase, we have <a==A, 6'==J9, ~"=w- <7,and the total
curvature is ~1+jB+C–M'. Hence thé excess over 180° of
the sum of the angles of a geodesic triangle is measnred bythe area of that portion of a unit aphere which corresponds tothe directions of the normals along the aides of the giventriangle.
The portion on the unit sphere corresponding to the area
enclosed by a geodesic returning mpon itself is half the sphere.For if the radius vector travel round so as to retum to the
point whence it set ont the extreme values of 6' and 6" are
eqnal, while <a bas increased by 2w. The measure of oir-
vature is therefore 8w or hatf the surface of the sphère.*
For Mme other interesting theorems, relative to the deformation ofsurfaces, see Mr. Jel!ett*s paper "On the Properties of InexteMÎNeSurfaces, ït'<tM«tt<MM<~<A<Royal ~fM&Academy, Vol. xxii. The theoryof mrfMes applicable to one another was the subject proposed by theFrench Academy as their Prise Question for t860, and the report of theCommission to which the décision was referred, gives reason to thinkthat the Memoirs sent in for compétition will, when published, add con-
siderably to what had been pïe')ioutly known on the subject.
( 312)
CHAPTER XII.
FAMILIE80F SURFACES.
879. LEftheequatïoasof&cMrve
iuclude n parameters, or undetermined constants: then it ia
evident that if n equations connecting these parameters be
given, the eurve is completely determined. If, however, onlya-1 relations between thé parameters be given, the équa-tions abovewritten may denote au-infinity of curves; and the
assemblageof a!I these curves constitates a surface whose
equationIs obtainedby eliminating the n parameters from the
given M+1équations viz. the M 1 rotations~and the two
equationsof the curve. Thus, for example, if the two equa*tions above written denote a variable curve~ the motion ofwhichis regniated by the conditionsthat it shall intersectM-11fixed directing curves, the problem is of the kind now under
coneideration. For by eliminating a', y, z between the two
equationsof the variable curvo and the two équations of anyone of the directing curves,we express the conditionthat thèse
two curvesshouldtintersect, and thus have one relation betweenthe n parameters. And having n 1 such relations we nnttthe equationof thé surfacegenerated, in the manner just stated.We had (Art. t09) a particular case of this problem.
Those sur&cesfor which the &rm of thé lunctions andis the same, are said to be of Me same ~ntt~, though the
equatious connecting the parameters may bo diferent Thuaif the motion of the same variable curve were regalated byscvcrat dînèrent sets of directing curves, aU the surfaces
generatod would be said to belong to the same family. ln
eeveralimportant casesthe equations of all surfaces belongingto the same family can be mdnded in one equation iavolving
FAMIME8 0F 8UMACE8. 818
oneor more arbitrary functtona the equationof any individual
surfaceof thé family being then got by particnlarizing the formof the functions. If we eliminate the arbitrary fanctionsbydifferentiation,we get a partial dîcerent!al equation, commonto all surfaces of the family, whieh ordimarilyia the expressionof Mme geometrical property common to all eur&oeaof the
family, and which !eadamore dtrectty than tho functionalequa-tion to the solutionofeorneclasBesof probleme.
380. The aimpteatcase ta when the equationsof thé variable
cnrve inelude but two coastanta.* Solving in turn for eachof
these conatants, we can throw the two given equations intothe form M=:c,,o='c,; where u and v are known &mct!onaof
a;,y, .e. In order that this curve may generate a aaï&oewe
must be given one relation connectingo,, c~ which will be ofthe form o,'=~(c,); whence putting for and o, their values,we aee that, whatever be the equationof connection,the equa-tion of the surfacegenerated must be ofthé form M= (v).
We can a!so în this case readily obtain the partial dMe-rential equation which must he satisfiedby atl Burfacesof the
family. For if <7=0 represents any such surface, U can onlydMerby a constant multiplier from « (v). Hence we have
\0= M (v),and dinerent!at!ng
with two 6!mHtrequations for the differentialswith respect to
y and <. EMm!natingthen and ~'(e), we get the reqniredpartial dî&rentMtlequationin the form of a determinant
If there were but one eoMtant the elimination of it would givethé equation of a defhMtesurface, not of a family of surfaces.
fAMtUES UF SURFACES.3M
In this case u and v are aupposed to be known funetions of the
co-ordinates; and thé equation just written establishes a reiation
< «'0'~ dUof the firat degree between
-y-, ,7 ?'
If the equation of the surface were written !n tho form
we 8bould have dU. 1, -P dU<(.B,~)~0; we ahouldh&ve ~=1, 1
where p and q have thé usual signification, and the partialdifferential equation of thé family ia of the jbrm ~+~=J?,
where $, are known funetions of the co-ordinates. And
conversely thé integral of such a partial differential equation,whieh (see Boole's .Dt~ereMtM~~tM~M, p. 322) is of the form
«==~ (e), geometncatly représenta a surface which can be gene-rated by the motion of a curve whose equations are of theform N==c,, e=c,.
The partial diSerential equation afforde thé readiest test
whether a given surface belonge to any asaigned family. We
bave only to give to f~, their values derived from the
cq~tion of the given surface, which values must identically
satiafy the partial differential equation of tho family if the
surface belong to that family.
881. If it be required to determine a particular surface ofa given family «=~(e), by thé condition that the surface shall
pass through a given curve, the form of the fonction in this
case can be found by writing down thé equations M=c~ <"=€and eliminating x, y, between theae equations aad those of
thé fixed curve, when we find a relation between c, and c~or between u and <~which is tho equation of the requiredsurface. The geometrical interpretation of this prooeas is that
we direct the motion of a variable curve «=!< e=i< by the
condition that it shall move so as always to intersect the givenfixed curve. All the points of the latter are there&M pointson thé surface generated.
If it be required to nnd a surface 'of thé family «==~(e)which ahaU envelope a given surface, we know that at every
point of the curve of contact 0,, 0~ 0,, &c. have the aame
value for thé fixed surface and for that which envelopes it.
815fAMtMES0FSURFACES.
If then in the partial differentialequation of the givenfamily,we substitntofor 0~, their valuesderivedfrom the equa.tion of tbe fixed surface, we get an equationwhieh will besatisfiedfor every point of the curve of contact, and whichthereforecombinedwith the equationof thé fixedsurfacedeter-
mines that curve. Thé problem is thereforereduced to thatconsideredin the first part of thia article; namety, to descnbea surfaceof the given familythrougha givencurve. All thia
theory will be better anderstoodfrom the followingexamplesof important familles of surfacesbelongingto thé claas here
coMtdered;viz.whoeeequationscan be expremedin the form
<t a (c).
882. C~N~fMf!?jSM~cM. A cylindricalsurface te gene-rated by the motion of a right Une, which remains always
parallel to itaelf. Now the equations of a right Ime ineludefour independentconstants; if then the directionof the rightlino be given, this determinestwo of the constants,and there
remainbut two undetermined. The family of cylindricalsur-facesbelongsto the classconsideredin the last two articles.
Thus if thé equationsof a right line be given in the form
<c=h+p, y'=mj!+~ Zand M which determinethe direcdons
of tho right Une are supposedto be given; and if thé motionof the right Unebe regulated by any condition(suchaa thatit shall movealonga certain fised eurve, or envelopea certainfixed sur&ce) this establishesa relation between p and q, and
the equation of the surfacecornesout in the form
FAMtUES0F 8MPACE8.816
equation of Art. 880, weMe that the partial dMeïentM eqa&-tion of cylindricalsur&cesM
or (Ex. 3, p. 26) P; cM<t+ C~ co8j8+ P~ cos'y'='0, where x, ~9,yare the d!rect!on-coa!nes of the generating Une. Rememberingthat C~, C~ are proportional to thé d!rect!on-co9lnea of the
normal to the surface, it ia obviowa that tbe geometrical mean-
ing of Uns equation ia that the tangent plaue to the surface
is always parallel to the direction of the generating Une.
Et. 1. To &id the equation of the cyRadet who<eedges are parallel to
.)t=&,y<M!,MtdwMehpMSMthroughthe plane cutv9it'=0t ~(i):,y)-0.~M. ~(~-&, y-MM)=0.
Ex. 2. To 6mdthe eqnation of the cylinder whose aides are parallel tothe mtarMctton of <M! + «:, <t'.);+ fy + <<, and which pMMBthroughthe inteMeetKm of ex + f3y+ <, J''(jr, y, <)s 0. Solve for a, y, <between the equations ax + + o; = «, «'< t fy + c'<= e, « + t < <<and substitute the rtautting values in .F(.)', y,<)0.
Ex. 3. To &td the equation of a cylinder, thé direetion-~osineeofwhoae
edges Me M, n, and whieh passes through the curve !7'= 0, 0. The
elimination may be conveniently performed a: foltow< If <e',y', <*be the
co~rdinatet of the point where any edge meeta the directing curve y, e
thme dany pointon theede we have g-W Y-$( 0-geCallingthose of any point on thé edge, we
hâve= !Ll~ *JL.. CaUing
the common value of these AtactioM C, we hâve
Substitnte theae values in the equations t~° 0, F'= 0, which <*y'z'must
eatisfl and between the two resulting equations eliminate the unlmown C,the reauIt will be the equation of the cylinder.
Ex. 4. To Cad the cytinder, the direction-eoainee of whose edgea are
m, n, and whieh envelopet the quadric ~z* ~y* t Ci~1. From the
partM dtfbtentM equation, the carve of contact u the intersection of the
quadrie with ~h + Bmy Chz = 0. Proceeding then M in the last examplethe eou~on of the cylinder ia found to he
383. CbKMoJ;S'M<~MM.These are generated by the motionof a right line which constantly pames through a fixed point.
Expreseingthat the co-ordinatesof this point satisfy the eqna-
FAM!UE8 OF SOBFACES. 317
tions of the right line, we have two retatIoMconnecting thefour constantsin the general equationsof a right line. In thiscase therefore the equations of the generating eurve containbut twoundeterminedconstants,and the problemie of the kinddtBcumedArt. 380.
Let the equationsof the generatingUnebe
where a, <yare the known co-ordinatesof the vertex of the
cône, and l, M,Mare proportionalto the direction-cosinesof the
generating line; and where the equations,though apparentlycontainingthree undetermined constants, actuallycontain onlytwo, Bineewo are only concernedwith the ratios of the quan-titlea Nt, M.
Writing the equations then in the ~onn
we aee that the conditions of the proMem muet eatabttsh a4M
relaûon between and and that the équationof the côneN M
muât be of thé&rm ~=~S-7 IJ-"f
It Measy to aee that this Méquivalentto aaytBgthat the
equation of the cône must be a homogeneousfunction of the
three quantitiesa: –a,y <!-'y; as maya!jMbe Beend!rectlyfrom the comideratton that the condïtionaof the proMemmuâtestahiNh a retat!on between the oirectiom-cosmeaof the gene-
ra.tor that thèse cosmeebemg.,M,–f.
&c' any equation
expresNngaud~a relation !sa homogeneousfunctionof Mt,M,and iheK&K of a:–a, y– ~-7< which are proportionaito m, n.
When the vertex of the cône!s the origin, tta eqna~on !a
of the torm -==<~('J or, m other worde,Ma homogeneons
functton of a;, y, <The partM diSerential equation ia found by puttmg
818 FAMtMES0F SURFACES.
«=– os* in the equation of Art. 880, and when~–y <y
>
deared of &actione N
Thia equation evidently expresses that the tangent plane at
any point of the surfacemust always pass through the fixed
point a~Qy.We have already given in p. 86 the method of formingthe
equation of the cône standing on a given curve and p. 190
the methodof formingthe equationof the conewhich envelopeaa given surface.
384. OMMMiMSurface8. Thèse are generated by themotionof a line which ahvays intersecte a fixed axis and remains
parallel to a fixed plane. These two conditions leave two ofthe constantsin thé equations of thé line undetermined,so thatthèse sur&cesare of thé class consideredArt. 880. If the axis!s the intersectionof the planes a, j8, and the generator is tobe parallel to the plane'y; the equatiousof the generator are
<t=c, 'y=< and the general equation of conoidat surfaces
Mobvioastya == (*y).*
The partial dinerentialequationis (Art. 380)
In Hkemannerthe equationof anymr&eegeneratedby the motion
of<tine meetm~twofb:edImet<t/9, mustbe ofthetMnt ~(~)'
FAMHJE9 OP SURFACES. 319
This eqaa.tton may bo derived direcdy by expreaamg that
the tangent plane at any point on thé surface contains tho gene-rator the tangent plane, therefore, the plane drawn throughthe point on the surface, parallel to the directing plane, and
the plane e'aj8' joining the same point to the axis, have
a common line of intersection. The terms of the determinant
just written are the coeniciente of x, y, <! in thé equationa of
thèse three planes.In practice we are almost exclusively concerned with right
conoids; that is, where the fixed axia ia perpendicnlar to the
directing plane. If that axis be taken as the axis of f, and
the plane for plane of ;cy, the tunctional equation is y = a~ (e),
and the rtial d!œ .al equation ismdU
+ y dU 0and thé partial dinepentia! equadon isa: ,-+y
= 0.
Thé lines of grcatcst slope (Art. 870) are in this case alwaya
projected Into circles. For in virtue of the partial differential
equation just written, the equation of Art. 870,
trMM&mna itself into .<&:+y<~=0, which represents a senea
of concentric circles. The same thing i8 evident geometricallyfor the lines of level are the generatora of the System; and
these being projected into a sénés of radil all passing throughthe origin, are eut orthogonally by a series of concentric circtea.
<
Ex. t. To Snd thé equation of the right conoid pMsict; through the
axis ofz and throngh a p!ane curve, who<e equaûoM are x=a, J*(y, )!)*'0.
EUmiMtmg then f, y, < between theae equations and y c e,<, < = < we
get F(c,a, <t) 0; or the requited equation faF(a:, z) ° 0.
WaUM'soon<w!nnemfawhen the Hxedcm've Ma cMe[;.)' a, y*<<*c ~].Its equation is therefore oy + <eV° f'.t*.
Eï, 2. Let thé direotin~ curve be a heim, the ftxed line being the axisof the cylinder on which the helix K traced. The equation is that givenBx. p. 273. This surface is often presented to the eye, being thatformed by the under surface of a spiral staitcaM.
385. jS't<y/sceaof ~epo&fttOK. The fondamental property of
a surface of revolution is that ita section perpendicniar to ita
FAMIUE8 OP SUBfACES.820
axis muet always cornât of one or more c!rcles whoaecentresare on the asia. Sach a mr&ce may therefore be conceivedM generated by a circle of YMiaNe radius whose centremoves along a fixed right line or axia, and whose plane is
perpeadtcular to that axis. If the equations of the axis beie–at v'j8 .v== == the gener&ttogcticle in any posi-
tion may be representedas the intersection of the plane per-pendîcnlar to the axis &c+M:y+oe=*c,,with the spherewhosecentre ia any Sxedpointon the axis
These equations contain bnt two undetermined constants; the
problem therefore is of the class considered(Art. 880) <mdthe
equation of the surfacemuetbe of the form
When the axis of e is the axis of révolution we maytake the
origin as the point ct~, <mdthe equationbecomes
The partial differentialequation is found by the formula of
Art. 380 to be
The partial differentialéquation expresses that the normal
always meets the axis of revolution. For if we wlahto ex-
prem the conditionthat the two lines
MMtMES 0F 8UMACM. 82t
T
should InteMCCt;we may write the commonvalue of the equalfractions in eMh c<Me, and Solving then for x, y, z, and
equating the values derived from the eqaatMmaof each line,we have
886. The equation of the surface generated by the revo-lution of a given curve round a given axia, ie found (Art. S81)by eliminating a',y, between
and the two equations of the curve; replacingthen u and v bytheir values. We havealready had an exampleof this (Ex. 3,
p. 85) fmdwe take M a further example to Snd the surfaco
generated by the revolutionof a circle [y= 0, (x a)t + e*==f*]round an axis in ita plane [the axis of <!]."
Putting <=«, a!'+y*=!C and eliminating between these
equations, and those of the cimle, we get
It ia ohvlonathat whena îs greater than r, that is to aay,whenthe revolving <arcledoes not meet the axta, neither can the
surface, which will be the form of an anchor ring, the spaceabout thé axis bemg empty. On thé other hand, when thé
revolving cu'de meets the axis, the segmentainto whichthe axis
divides the c!rclegenerate dtatmet sheets of the surface, inter-
secting in points on thé axie ~==~(~<~), which are nodal
points on thé em'&ce.The motions of thé anchor ring by planes parallel to the
axis are found by putting y ~constant in the precedingeqna-tion. The equationof the section may immediatelybe thrown
822 fAMtLtMOP SURFACES.
into the form M'<=constant,where and <S"represeat c!n!lea.The Beetionsare lemnMcateaof vanoas Mnds (aee6g., B~~e!'JP&MMOitn~, p. ~04). It Mgeometricallyevident, that as the
planeof Motionmores away 6'om the axis, it continuesto eutin two distinct ovals, until it touches the surface [~~a–fjwhen it enta in acurve havinga doublepoint ~BemonitK'sLem-
mBcate] after whichit meetsin a continuonscurve.
Ex. Verify<httt.)~t y*-t -'3~ ° iaa «M&eeoffevotatton.~M<.TheMMofrevolutioni< c y e.
387. The familles of snr&cea which have been consideredare the most interesting of those whose equations can be ex-
pressed in the form M'== (v). We now proceed to the case
whenthe equationsof the generating curve inchtde more thantwo parameters. By the help of the equations connectingtheaeparameters, wecan, in terms of any one of them, expressall the rest; and thusput the equationsof the generatingeurveintothe form
The equationof the surfacegenerated Mobtained by elimi-
nating o between these equations; and, as hae been aireadyatated, all sur&ceaare said to be of the aame familyfor whichthe form of thé funetionsF and f is the Mme, whatever be thé
form of the omettons &c. Bat aiBceevidently theelimination cannot be ef~cted until some defmite form haabeen asaigned to the ~anctiona &o. it is not genetatly
poaaîMeto forma amgle fanettond equation mdadimg aUsur.
faces of the same ûmuty: and we can only represent them,as above written, by a pair of equationa from wMch thereMma!naa constantto be eliminated. We can however elimi-nate the arbttrary fanctions by differentiation and obtain a
partial aifferentialequation,commonto aUaar&ceaof the same
family the order of that equation being, as we ahaMpreaently
prove,eqnal to the numberof arbitrary fanctïoms &c.It ia to be remarkedhowever that in general the order of
thé partial diferential equation obtained by the elimination of
a number of arbitraryfonctionsSroman equation is higher than
fAMtMZS0F SCM'ACZS. 829
the number of fanct!onaetiminated. Thus if an équation in-
cMe two arbitrary fanctions and if we dIneMnttatewith
respect to x and y which we take as independentvariables,thé dicerentiats combined with the original equationform a
aystem of three equations containing four unknown fancûoM
The second dtnerentiatton (twice with regardto x, twice with regard to y, and with regard to x and ~)gives ua three additional equatMns; but then from the syatemof six equatioM it is not generaUypossible to eliminatethesix quantities We must thereforepro-ceed to a third differentiationbefore the elimination can beeNected. It is easy to see, in like manner, that to elimmaten arbitrary fancdons we muet differentiateSht 1 tunes. Thereason why, in thé présent case, the order of the diSerent!al
equationMlésa,is that the functionseliminatedare aMfuactîona
0/*<i&esamequantity.
888. In order to show this it !s convenient to considerfirstthé special case, where a family of surfacescambe expressedby a single fnnct!onalequation. ThM will happen when it is
possibleby combining the equations of the generating cnrveto separate one of the constants so as to throw the equationsintothe&rm«==o,; ~'('e<y)~CttC,c,)='0. Then express-ing,by means of thé equationsof condition,the otherconstantsin terms of c,)the reault of eliminationis plainlyof theform
Nowif, as before,wedenoteby the differentialwith reapectto a!of the equation of the surface,and by F,, the differentialonthe MppomtMnthat u iaconstant,we bave
Now in these equations, the derived functions< &c. oniyY2
FAMÏHES0F SURFACES.824
which contains only thé original OmctMM &c. If wewrite this equation F=.0, wo can form &om it in like mannerthe equation
which atill contains no arbitrary functiona but thé original&c.,but whichcontainsthe seconddifferentialcoen!cients
of U, these entering into P,, From the equation last&~mdwe can in like manner form another, and so on; andfrom the sertes of equationa thus obtained (the last being of
the K**order of dfferentiation)we can eliminatethe a fhncttons&c.
If we omit the last of these équation~ we can eliminateallbut one of the arbitrary functions, snd accordmgto our choice
of the fonctionto be retained, can obtam ? dinerent equationaof the order M-1, each containing one arbitrary fonction.These are the first ïntegrab of thé final di~rentia! equation
of the M"'order. In like manner we canibnn eqaa-
tions of thé secondorder, eaeb containingtwo arbitrary fanc-
tions,and ao on.
889. If we take a: and y as the independentvariables, and
as nsaat write d<t='p<!a!+~, <~=M&c+«~, &c., the procomof forming these equations may be more conveniently atated
as follows: "Take the total dtSerentM of the given equationonthe suppositionthat u Mconstant,
FAMtUES OF SURFACES. 325
JTT*and the result of eliminating -j-
&omthefte two equationsts
the same aa the reault of eUmmatimfm betweenthe eqn~onB
It is convenient in practioeto choosefor one of the equations
representingthe generating curvo, its projection on the planeof a~ then since this equation does not contain e, the valueof m derivedfrom it will not contain p or aad the firat
diSiaremtMdequation will be of the form
B being a1soa function not containingp or g'. The only terme
then containingf, N)or t in tho seconddigerential equationarethose derived from differentiafingjp+~t, and that equationwill be of the fbnn
where jSmay contain ?, y, e, p, q, but not r, a, or <. If nowwe had only two thnettons to eliminate,we ehouldsolveforthese constants from the original fonctional equation of the
anrface,andfrom p + qm= B; and then snbstttntmgtheaevaluesin mand in the formof the final seconddifferentialequationwouldanUremain
where m' and jS' might contam x, y, < In like mannerif we had three functions to eliminate, and if we denote the
partial differentialeof <! of the third order by <!t,~8,'y, the
partial dM~rential equation would be of the form
FAMtUM0F aUNPACES.826
And so on for higher orden. This theory wiHbu illuatrated
by the examp!eswMohMov.
890. ~«<~MM~Mfa~ ~e< jXM'a&~<o<!/.Ke<?plane.This !a a family of sar&coswMch ineludes conoidsM a pM-ticular case. Let us in the 6nt place take the Ëxod phmefor the plane of xy. Then the equattom of the generatingline me of the fom a~c,, y~e~c+c~ The 6mcttMM~equa-tion of the surface ia got by sahstttatuagin the latter equationfor c,, (e), and for c~ (e). Since in forming the partialâMbrentM eqa~on we are to regard e aa constant,we mayas weU leave the equationa in the form es*e~ y='c~E-Thèse give us
AocoKHagas we eliminate o. or c,, thèse equations give na
p-t<!c,=0, Ft+2y=~ There are therefore two equationsof the Rmtorder, each containingonearbitrary fnncdon,vis.
To eUmhate completely arbitrary Atnctîons, tM~rentt&te
jp-t-~t~o, rememberingthat mneeM=e, tt Mto be regardedaaconstMtt,whenw6g<!t
and eliminatingm ty means ef p+ <jw=0, the required eq<Mt-tion Is
Nextlat&egeaetatmg~MbepoKtMtooiC+~+ei!;tta equatiom are
and thé <&)nct!oMlequationof the &auly of tfor&ceais got bywriting foro, and c,, foBCtMCBof oa: + + <?. DMEMentt&tIcg,we have
The eqa&tMomgot by eHmumtutgone arbitrary faactiomaretherefore
FAMtUM 0F SURFACES. m
891. This equation may atM be arrived at by expMMtngthat thé tangent planes at two points on thé same generatorintersect, as they evidently must, on that generator. Let
a, /9)y be the raamng co-ord!nate«,a', y, e those of the pointof contact; then any generator is the intersection of the tan.
gent plane
Nowif wepaesto the line of intersectionof thu tangent planewith a consecutiveplane, a, <yremain the same, while
a:, y, e,p, g vary. DiRerentia~ngthe equation of the tangentplane,we have
?2. t6'M)~MMj~MM~M)!'6~ KMM<oAtC~<tMe<a ~!M<?axis.Thia class aleo ineludes the family of conoids. In the nrat
place let the fixed axis be the axis of e; then the equationsof the generating Uneare of the formy=c,a:, <!==c,a:+e,;andthe equation of the family of aarËMeeMgot by writing in the
FAMtMES0F 60BFACES.828
latter equation for o. and c,, arbitrary AmetiolMof Diffe-1#
rentiating, we have Mta'c,+)Mg'ac,) whence
Difforentiatingagain, we bave f+8Mt+<M*=*0, and putting
for M its vàhie =c,=' thé MqMredd!Sereniud equation is0
This equation may aiso be obtained by ejq~ressingthat two
consécutive tangent planes întemect in a generator. Aa, in
Art. 891, we have for the intersectionof two consecadve Km-
e'ent planes
But anygcnerator Mes!nthe ptane e~=~.c, or (a–.e)~=*(~-y)ic.EIinum&tmgtherefore
B~t-~=~=~. There6)re,Mbe&)re,ra:*+8&Ey+~'=0.<<a*a a? f
More generaHylet the line pass through a &ced axut c<t8,where a='<!a!+~+o&-t-<< ~=a'a:+&'y+p'e+<y. Then the
equationsof the generating line are <( c~, y =o~ + c~and the
eqa&tMnof the &auly of sm'&ceaMy=.a!~~+~
D!
tentmttBg)we hâve
Differentiatingagaia, we have f-t-Z~M+<)?''=(), and puttingin form from the last equation, the required partial diSerentta!
equationis
893. If the equation of a family of snr&ces oontain n
arbitrary jRmctionaof the same quantity, and if it be required
PAMtMEN0F 8UM'ACE8. 889
to determinea aurface of the family which shallpMs thronghMfixed curves, we write down the équationsof the generatingcarve M=~, F(a:,y, e, c~,cl &c.)=0, anAexpreaing that the
generating curve meets each of the fixed OHrvef),we hâve aM~dent number of equations to eliminate c,, c,, &c. Thusto &td a M!'&c&of the family a:-ty~ (<)-t (<)'=0wMdi ehaU
pMs through the &ted cnrves at JF*(i<<)=<0 y = a,J~(a:,z)'s 0. The equationsof the generatinglinebeing e ~=c,,a!=*ye,+~ we have, by substitution,
TheKMith appatendyof the eighthdepee,butMKMhaNeintotwoceno!d<distinguishedby p~u)~ the Mdittththe sameo)'opposite~p)B& the lastequttdon.
894. We have nowseen that when the equationof a &milyof sar&ceecontains a number of arbitrary fanctionsof thé Mme
quantity, it M conveniecft,in forming the partial ~fBM'en&d
equation, to substitutefor the equation of the surface,the two
equations of thé generating curve. It îs eaay to aee thenthat this prooesa ia equally applicable when the family of
Mr&cea cannot be oxpressed by a a!ng!efamcëontjequation.The arbitrary funetionswhich enter into thé equations(Art.387)are all functionsof thé same quantity, though the expressionof
that quantity in termeof the co-ordinatesis unknown. If then
fAMtMES <? SC~ACM.880
tM~rentiatiag that quantity gives a~ "md~c,we cm eMminatethe unknown quantityM, between thé total dinMentiahof thétwo equationsof the generating curve, and so obtainthe partialdMRerentiatequation required. Im practioeit is convenientto
choose for one of the equations of the generating ourve,its
prejection on the plane xy.For example,let it be required to fmdthe generalequation
of rnled aar6Mes;that is to say, of sar&cesgenerated by themotion of a right Une. The eq~iatioMof the generating line
aMz=e,a!-(-c,, ~'cc~c+c,, and the family of surfacesis ex-
pressed by snbBtttntmgfor c~ c,, arbitrary <uBct!onaof c,.DMbMnttatmg,we have ~+M~=~, w=e,. Dl&rentiatmgthe first of these equations,m being proved to be constantbythe second, wehave <'+8aM-t<M'c.O. As this equationstillmdttdes m or <~ the expresdon for which, in. terms of the
co-ordinatesis unknown,we muet d!SereNtiateagain, whenwehave <t+8~3m-<-8'ym'+&M''='0,where a, /3, y S are the'thirddifferentialcoeCdents. EMm!natmgm betweenthe cuMcand
quaftraticjust found,we have thé requiredpartial dMerential
equation. It ovidentlyresolves itself into the two linear équa-tions of the third order got by substituting in turn for M inthe euMcthé two roots of the quadratio.
This equationmight be got geometricallyby oxpresmngthat
the tangent planes at three consécutivepoints on a generator
pass throngh that generator. The equation<&<&!+~ is
a relation between1) wh!ehare proportionalto the direc-
tion-cosinesof a tangent plane, while<&e,< <&are proportionalto the direction-cosinesof any line m that planepassingthroughthe point of contact. If then wepaasto a secondtangentplane,through a consécutivepoint on the same line, we are to make
p, vary whilethe mutual ratios of & <~ <&remain constant.This gives ~&+2«&eo~+<t<=0. To pass to a third tan-
gent plane, we dineron~ateagain, regardingdx <~ constantiand thus have <«&+8~<b~<~+8~<~<~+&=0. E!imi-
nating die dy betweenthe last two equations,we have the
eame equation as before.
The fint integrals of this equation are found,as explained(Art. $88), by omitting the last equationand eliminating all
fAMÏMEa0F SUBFACES. Ml
bat one of the constants. Thua we hâve the equation
p+<M~=c,, from wh!ch it appeara that one cf the mtegtab is
p+)K~*s~(<M),wheromia one of the roota of f+2<m+<<N'=c0..Thé other two ËKt mtegralaare
Thé three second integtata are got by eliminatingm &0!n
any pair of theae equations.
396. J5t<M&pM.If the équation of a Bar&ceiadade «
ptHf&meteMcoïmectedby M-] relationB,we can in tefms of
any one express a!l the Kst, and throw the equation into
the formY:fL_ s i_~.ni_v o__~tf < t e~~
EËminating 0 between this equation and -='0, we find the
envelopeof aU the surfacesobtainedby giving dînèrentvalues
toc. TheenvdopeaaofbnndaresatdtobeofthesamefamilyM long ae the form of the fonctionF renMunBthe same,no matter how thé forme of the ftmctMns && vary.
<H!*The carve of mtersecttonof the given surface with is the
<<tM!C<~&<&(aeo p. 882) or line of intersection of two con-secutive sariacea of the system. Consideringthe charactenstM
aBa moveable cnrve from thé two équationsof whichc Mtobe eliminated, it is evident that thé problem of envelopesisincinded in that discassed,Art. 887, &c. If the nmetionJ'*
~Etcontaina arbitrary onctions &c.,then smce contams
&c., it would seem, accordingto the theory prev!oadyexplained, that the partial diSerential equation of the familyonght to be of the 2tt*"order. But OBrexaminmgthe manner
in whieh these functîonsenter, it Measy to see that the orderredaces to the ?' In fact, differentiatingthe equation<tc:~we get
FAMtUES0F 8UNFACB8.332
are thé d!nëren<ia!son the apposition that c !a constant, these
quantities only containthé origmal ~anet!ons and not the
derived From this pair of equations we can form
another, as in Art. 894, and où on, until we come to the M"*
order, when, as easily appesm from what &Uows,we have
equations enough to eliminate a!l the parametem.
3M. We neednot considerthe case whenthe givenequationcoNtainsbut one parameter,Nneethe eliminationofthMbetween
the equation and ita difFerentialgives rise to the equation of
a definiteaarface and not of a family of sor&ces. Let the
equation then contain two parametera <~ connectedby an
equation giving b aa a fomctKmof o, then between the three
equations<=j~~==J~, ~=~ we can eliminate <t, and the
form ofthe result ia ev!dently/(!c,y, ~,p) g)= 0.For example, let us examine the envelope of a aphereof
Sxedradius, whosecentre movesalong any plane curve in the
plane of Thia ia a particular case of the general dass oftubatar surfaceswhichwe ahallconsiderpresently.
Nowthé equationof encha sphèrebeing
and the conditionsof the problem&aa!gnmga locus alongwhtchthe point <~ is to move, and thereforedetenninmg/3 in termsof <[,the equation of the envelopeis got by eliminating a
between
S!nce the eliminationcannot be effected until the form of the
function is aBMgned,the familyof surfacescan only be ex-
pressed by the combinationof two eqmat!oMjust written.
We might aiso obtain thèse equationsby expressingthat the
surface is generated by a jExedcircle, whmh moves so tbat!t9plane shan be alwaysperpendicalar to thé path along whichits centre moves. For thé equation of the tangent to the
locusof e~ is
FAMtMES0FaUBFACES. 888
ag already obtained. To obtain the partial differentialequa-tion, differentiatethe equation of the ephere,regarding a, as
constant,when we have if <t+jpi!!=0,y ~8+ = 0. Solvingfor a'-a, y-~8 and substituting in the equation of the sphere,thé required equation!s
We might hâve at once obtained thîs equation as thé geo-metrical expression of the fact that the length of the normalis constant and equal to r, as it obviouatyM.
397. Before proceeding further we wish to showhow the
arbitrary fanctions which occur in the equation of a familyof envelopes can be determined by the conditionsthat the
surfacein question passes through given curves. The tangentline to one of thé given eurvesat any point of courseMeainthe tangent plane to the required sur&ce; but Bmcethe en-
velopingsurface bas at any pomt the same tangent plane as
thé enveloped eor&ce which passes throngh that point, itfollowsthat each of the given oarvea at every point of ittouchesthe envelopedsurfacewhichpaMesthrongh that point.If then the equation of the envelopedsurfacehe
the envelopeof thta aor&cecan be made to pass thjoagh M-1i
given curves; for by expressîngthat thé surfacewhose equa-tion has been just written touches eaeh of thé given corves,we obtain M–l 1relationsbetween the constante c,, c,, &c.,which combined with the two equationsof the characterisdcenable es to eliminate theM constants. For example, the
family of Mr&cea discasMd in the last article contains buttwo constante and one arbitrary fnnction, and can thereforebe made to pass through one given carre. Let it then be
required to find an envelopeof the epheM
~UMtHESor SURFACES.834
pomte of intersection of this line with the sphete beiag gt~ea
by the quadratic
the conditionthat the lineehouldtoach the sphèreis
We see thus that the locusof the centres of apherestouchingthe given line is an eU!pM. The envelope required then iaa kind of eHtpt!Ct~anchor ring, whose equation M got byeliminatingft) ~3between
6'om wtuchlast two equationswehave
(1+ M*)j3(a;-a) =<t(y -~8).
Thé resottis a sur&ceof the eighth degree.
398. Again, let it be requiredto determinethe arbitraryfanction so that the envelope sar&ce may aho envelope a
given surface. At any point of contact of the required sar-&ce with the fixed surface a =y(a', ~), thé moveable surface
<!=jP'(a:,y,c,,c,,&c.) which paMes throngh that point, has
also the Mmetangent plane as the fixed surface. The values
then of p and q derived6'om the equations of the fixedsurfaceand of the moveablesurfacemnst be the aame. Thus wehave
~==J~, ~=~,) and if between thèse equations and the two
equations e'='~ <!==~which are a&tMnedfor the point of
contact, we eliminate x, y, <, the reault will give a relationbetween the parameten. The envelope may thua be madeto envelope as many fixed saï&ces as there are arbitrarytunctions in the equation. Thus, for example, !et it be re-
quired to determine a tubular surface of the kind dieenssed
(Art. 897), whichahall touch the aphereit''+~*+ ~'=~B'. Thia
surface must then touch (~-M)'+(y–~+z''=~. We have
there&re =* ) *= 'L-condidonswhich !mplye = 0,c
= ~11" or ~.t:='<ty. EUminating !Cand y hy the help of
PAMtMJEOOf 80BFACBS. 8M
thèse equations,between the equation of the fixed and move-able sphere, we get 4(<t*+~')~*=(~+~'+/3~ Thia
gives a quadratic for a*+~8*,whose roota are (~if)'; showingthat the centre of the moveableapheremoveson one or otherof two circles, the radins being either JBif. The surface
required M therefore one or other of two anchor rings, the
oponingof the rmgecorrespondingto thé valuesjust assigned.
399. We add one or two more exampteaof families of en-
velopeawhoseequationainclude but one arbitraryfunction. To&i(t the envelope of a right cone whoaeaxis Mparallel to théaxis of < and whose vertex movea along any assigned curvein the plane of a:y. Let the equation of the cone in ita
original position be <=M'(.~+~); then if thé vertex bemoved to the point a, the equation of the cone becomes
e'=M*((.e-e!)'+(y-~3)'}, and if we are given a curve
along which the vertex moves, ~8 is given in terms of a.
IM~renUat!ng we have ~!==m*(a;-ct), ~ec*m*(y–/9); and
eliminating we have j)'==?'. Thia equation expressesthat the tangent plane to the surface makes a constant anglewith thé plane of a~, as !s evident from the modeof generation.It can easily bo deduced hence that the area of any portionof the surface !s in a constant ratio to tts projection on thé
plane of xy.
400. The &mtnea of surfaces, considered(Arts. 896, 399),are both ineluded in the following "To findthe envelopeof asurface of any form which moveawithout rotation, ita motion
being directed by a cnrve along which any given point of thesurfacemoves." Let the equation of the surfacein its original
position be z=.F(a:,y), then if it be movedwithout turning80 that the point originally at the origin ahall pa~Mto the
position o~-y, the equation of the surface will evidently be
t!y.=J!'(<c–6t,y–~3). If we are given a cnrve along whichthe point a~3'yIs to move, we can express a, ~3in terms of -y,and the problem ia one of the ctass to be consideredin thenext article, where the equation of the envelopeineludes two
arbitrary funetions. Let it be given howeverthat the directing
FAMÏMM 0F 8CMACE8.?6
curve & <&'<HMton a certain &!M<paMf~~ce,then, of the two
equationsof the <)reet!ngcurve, one is known and only one
arbitrary, eo that the equation of the envelopeindudes but
one arbitrary function. Thus if we MStune an arbitraryfonctionof a, the equation of the &ted surfacegives 'y M a
knownfunctionof a, /3. It is eaayto seehowto findthe partialdM~fent!aIequationin this eaae. Betweenthe three equations
If then the equationof the surfacealong whiehft~-yis to movebe r (a,~3,y)= 0, the required partial dMSerentitJequationM
The three functionej~ are evidently connectedby the
te!att<m<ry'=~'+g<~It ia easy to see that the pMrtMdifferentialequationjust
found is the expreamonof the fact that the tangent plane at
any point on the envelope, ia parallel to that at the corM-
apondingpoint on the ong!nat stu'&ce.
Ex. To and thé partial differentialequationof the envelopeof aapheMof coMtantMdhMwhoMentre movesalongany cm'vetracedont&MdeqnatapheM
FAMtUEa 0F SURFACES. 887
z
401. We now proceedto investigatethe formof the partialdifferential cquation of the anvetape,whenthe equation of thc
moveable surface containsthree constanteconnectcdby tworolattona. If the equationof the surfacebe <!==JF(a',y, < c),then we have ~!=~, g'=~. Dt&rentiatmg again, as in
Art. 889, we have
The fcnctiûM 2~ contain a, &,c, for whieh we are
to aabstttate their values in tenus of a:,y, < derived from
eotvmg the preceding three equations, when we obtain an eqn&-tion of the form
402. The following examples Me among the most importantof the cases where the equation inclades three parameters.
Decei'opo&&&<t~MM. These are the envelope of the plane
=<t!K+ + e, where &r 6 and c we may wnte (a) and (a).
DM~rentIatmg we have~=<t, ~==&,whence ~=~(~). Anysurface therefore is a developable snr&ce if and q are con-
Dected hy a relation mdependent of a:, y, e. Thus the family
(Art. 899) for which p'+g'~M*, is a family of developablesurfaces. We have also –~a' S~*= (~)) which is the other
first integral of the final differential equation. This last is
got by diNerentiating again the equations p= a, g =&, when
we have )'+<w=='0, a+<m==0, and eliminating M, W–a'~O,which M the required equations. 1
1 owe to FrofeMor Boole my knowledge of the &et that when the
equation of the moveable aurfnee containa three parametera, thé partialdifferential equation Mof thé form stated above. He has kindty aUowed
me to consult, pre~ioM to Its publication, a memoir of his in whieh
thit theorem ie given.
PAMtMEa (JF SURFACES.338
By companng Arts. 264,281 !t appears that the con~tion
f<==<*is MtMed at every parabolic point on a surface. The
aame thing may be shown directly by transforming the eqnationf<=0 into a function of the dift'erential coefficients of
by the help of the relations
when the equation r< becomes identical with the equationof the Hess!an. We Bee now then that every point on a
developable is a parabolie point, aa is otherwise evident, for
mnce (Art. 298) the tangent plane &t any point meets the
surface in two coincident right lines, the two !nâex!onal
tangents at that point coincide. The Hessian of a develop-able must therefore always contain the equation of the surface
itaelf as a factor. The HeMiaa of any sot&ce bcing of the
degree 4H–8, that of a developable consists of thé surface
itseU, and a sur&ee of 3~–8 degree which we ahall calt
the Pro-Hessian. We may return to this subject herca~er.
403. ?~S«&!)'~«t/acM. Let it be required to Snd the
diB~rential equation of the envelope of a sphere of constant
radius, whose centre moves on any curve. We have, as in
Art. 400,
PAMtMES 0F SURFACES. 839
Z2
whieh denotes, Art. 281, that at any point on the required
envelope one of the two pnuctpat radil of e<tf?atNM!s equalto as is geometrically evident.
404. We shall briefly show what the form of the ditfe-
rential equation is whcn tho equation of thé surface whoM
envelope is sought contains four constants. Wo hâve, as
before, in addition to thé equation of the surface the three
equations F =~, '? =~, ('- -) (<- ~) (< ~,)'. Let u",for shortness, write the last equation pr==*er', and let us write
<!t-=~ ~=~, V-~=C', o-=Z); tlien, diffe-
rentiating pT=o- we bavee
(~ +J?w) -r+(C+Deet) p-2 (B+Cm) <r=0.
Substituting for <Mfrom thé equation o'+TM=0, and remember-
ing that'~T = we havo
in which equation we are to substitute for thé parametera im-
plicitly involved in it, their values derived from the preceding
equations. The equation is therefore of thé form
6[+3~M+3~'+Om'=~
where M and U are functions of a', y, s, p, q, r, s, t. In like
manner we can form the differential equation when thé equa-t!on of the moveable surface mctndes a greater number of
parameters.
40S. Having in the preceding articles explained how
pM~at differential equations are formed, we ahaH next ahow
how from a given partîal differential equation can be de-
rived another differential equation satianed by every charac-
teristic of the &m!ty of Sta&ces to which the given equation
belongs (see Monge, p. 58). ïn the first place, let the given
equation be of the first order; that M to say, of the fonn
j~(!B,y,e,p,g)=0. Now if thia equation belong to the en-
velope of a moveable surface,' it will be utisfied not only bythe envelope but also by thé moveable surface in any of Its
positions. Thia follows from the fact that the envelope touches
the moveable surface, and therefore that at thc point of contact
a:, y, s, p, q are thé same for both. Now if x, y, z be the
FAM!MES ()F SURFACES.340
co-ordinatesof any point on the characteristic,fince such a
point M the intemectiooof two conseeutivepositionsof themoveable surface, the equation y(a:, y, t!, p, $)=0 wHt be
satisded by thMe values of e, whetherp and hâve the
values derived from one position of the moveablesurfaceor
from the next consécutive. Consequently~if we differentiatethe given equation, regarding p and q as alonevanaMe, then
the pomts of the cbaracteristîcmuat a&tiaf~the equation
Or we might hâve stated the matter M follows: Let the
equation of the moveable surface be s = F(a:, y, a), where
the conatant: have all been expressed as fonctions of a single
pammeter a. Then (Art. 895) we have p = j~ (iC,y, a),
<~=J~(ic, y, a), whieh vaines of p and q may be saba~tnteA in
the given equation. Now the characteristic is expreMed by
combiuing with thé given equation tt9 differential with respectto a and a only entera into the given equation in consequenceof its entering into the values for p and q. Hence we have,
Now smcethé tangent lino to the charaetensttcat any pointof it, t!ee in the tangent plane to either of thé mr&ceswh!chintersect in that point, thé eqnattoc <&==~<&:+%t~is satMed,whetherp and q have the values derived from one positionofthé moveablesttrface or &omthe noxt consécutive. We hâve
therefore <&+-,= <~= 0. And combiningthis eqaationwith
that previoualyfound, wechtain the d~erenëal eqnat!onof the
çharacteristicJ~ ~<&:=0.Thua if the given eqmtttonbe of thé form .%+(~=.B,
the dMractenatMaattsËeathe equation JR~ <3(&!= 0, fromwhich equation combined w!th the given eqaat!onand with
<&=y<&!+~, can be dedaced jR&=&& ~=.B< The
Mader Maware (see Boote's jM~eMtM~~M~MM, p. 888)ofthe use made of thoae eqnaëona in integratmg thie chas of
eqnationa. In &ct, if the above eyatemof amnitaneoasequa-~ona integrated give w='< c=c,, theae are the equationaof
FAMtUtS 0F aUM'ACES. 341
the chartMtetisttc, or generating cnrve, in any of ita positions,while in order that wmay be constant whenever tt is eonstant,we muttt have M= (v).
Ex. Let the equationbe that comidered(Art. 396),vis.)!*(! }*)=t*,then any chaMetoMe Mt~tee the equation~y c ~th', wMchtadicttea
(Art.370) that the ehMMter!eUtia dwayea line of greatestdope on theMt&ce,as la~eometncaUyévident.
406. The equation just <baDdfor the chamctertsttc generally
Inctudea aad g, but we can eliminate these quantities by com-
bining with the equation jast found, the given partial di8e-rential equation and the equation <&!=~<&!+~. Thm~ in the
!Mtexample, from the equations z*(l-)-p't~')!=y', y<&=j~,we derive
The reader is aware that there are two classes of differential
equations of the first order, one derived from the equation of
a single surface, as, for instance, by thé elimination of anyconstant from an equation 0~=0, and its differential
An equation of this dass expresse: a relation between thedirection-cosmosof every tangent line drawn at any point onthe sur<ajoe.The other class is obtained by combining the
equationsof twosnr&ces,as, for instance,by eliminatingthreeconstants between the equations !7=0, F==0 and their dISë-rentials. An equationof tbis daœ expressesa relation satis&ed
by the direction-cosinesof the tangent to any of thé curveswhich the system J~ F represents for any value of thé con-stants. Thé equationsnow under considerationbelong to thelatter class. Thua the geometricalmeaning of the equationchosenfor the example is that the tangent to any of the curveadenoted by it, makes with the plane of a~ an angle whose
cosine is This property is true of every cirde in a vertical
plane whose radius is r; and the equationmight be obtained
by eliminatingthe constantsa, ~9,m, betweenthe equations
FAMtUM 0F SURFACES.342
407. The dîSerentiat equation found, es in the last article,is not only true for every characteristic of a family of sur&ces,but since each characteristic touches the cuspidal edge of thé
surface generated, the ratios (&e d~ t: <& are the same for
any characteristic and the corresponding cuspidal edge; and
consequently the equation now found is satisfied by the cuspidal
edge of every surface of the family under consideration. Thus
in the example chosen, the geometrical property expreMed bythe differential equation not only M true for a circle in a
vertical plane, but remams tme if the circle be wrapped on
any vertical cylmder and the cuspidal edge of the given
family of surfaces aiwaye belongs to the family of eurves thus
generated.
Prec!sely as a partial differential équation in p, y (expresa-
ing as it does a relation between thé directiom-coNnes of the
tangent plane), la tme as well for the envelope as for the par-ticular sur&ces enveloped; so the total differential equations here
considered are trae both for the cuspidat edge and the series
of characteriatica whieh that edge touches. The same thing
may ho stated otherwise as follows thé aystem of equations
P=0, ~-==00 which, when a is regarded as constant, represents
the characteristic, represents the cuspidal edge when a is an
unknown function of the variables to be eliminated by means
xof thé equation -,y='0. But evidently the equations P==0,
=0 hâve thé same d!Berentiats when a is considered as<M
vanaMe, subject to this condition, as if a were constant.
Thus, in the example of the last article, if in the equations
(.c-<!f)'+(y-~)*+~=~, (a:-<
FAMILIES OF 8CRFACBS. 343
vmy according to the ruies hère laid down. And we ahaUobtain the equationa of a curve satisfying thia diSercat!~
equation by giving any form we pteaae to ~(a) and then
eliminating between the equations
408. In like manner can be found thé differential equationof the characteristic, thé given equation being of the secondorder (see Monge, p. 74). In this case we can have two
consécutive surfaces, satis~ing the given differential equation,and touching each other all along their Une of intersection.
For instance, if we had a Mrface generated by a carve movingeo as to meet two nxed directing curves, we might conceive
a new surface generated by the Mme curve meeting two new
d!recdng curvcs, and if these latter directing curves touch the
former at thé points where the generating curve meets them,it is evident that the two surfaces touch along this line. In
the case supposed then the two surfaces bave !C,y,common along their line of intersection and can differ onlywith regard to r, <, t. Differentiate then the given dl&;rentiai
equation considenng these quantities alone variable, and let
It is convenient to insert here a remark made by Mr. M. Roberts,vix. that if in the équation of any surfaee we substitute for r, < + Xd~,
for y, y ~y* for ~t s +~<&tand thon form the discriminant with respectto X, the result will be the differential equation of the ouspidal edge of
any developable enveloping the given surface. In fact it evident (seeArt- 246) that the dhofiminant expresses tho condition that the tangentto the carve represented by it touches the given surface. Thua the generalequation of the cuspidal edge of developables cireunMcriMnga sphère is
In the latter form it M evident that the same equation is satisfied bya geodesic traced on any ocne whose vertex is thé origin. For if thécone be developed into a plane, thé geodemc wi)l become a right line,and if the distance of that Une from the origin be a, then thé area of the
triangle formed by joiaing any element <!<to the origin is half ;«/<, but
thia is evidently the proporty expressed by the ptcceding equation.
FAMtLtES0F SURFACES.844
the reault be JM5'+ jS&+ Tdtm0. But emcep andq are con-
<t!Hit&IongthM!me,wahave <?r<Xc+obd~=0,<&<&c+<&<~<=0.EUmm&dngtheo ob, dt, the requited equation for the chtt-
racteristicla
In the case of any of the equations of the second order,which we have ah'eady had, this equation would tara out a
permet square. When it does not so tum out, it breaks upinto two &ctom, which, if mtioDat, behmgto two independent,charactensttcs represented by aeparate equations; and if not,denote two branches of the same curve mtersectmg on the pointof the surface which we are considering.
409. la fttct when the motion of a surface is regulated bya smgte parameter (see Art. 290), the equation of its envelope,as we have aeen, contams only fanetions of a smgte quantity,and the differential equation belongs to the simpler speciea
just referred to. But if the motion of the surface he regulated
by two parameters, its contact with its envelope being not a
curve, but a point; then the equation of the envelope will
in general contain fonctions of two qnantities, and the dine-
rential equation will be of the more general form. As an
illustration of the occurrence of thé latter c]as9 of equations in
geometrical investigations, we take thé equation of the familyof sur&ces which bas one set of its Unes of curvature parallelto a fixed plane, y=mx. Putting <j~'=M<&;in the equationof Art. 280, the differential equation of the family is
As it does not enter into the plan of this treatise to treat ofthé integration of such equations, we refer to Honge, p. 161for a very interesting diaensmonof this equation. Onr objectbeing only t~ show how snch differential equations presentthemaetveain geometry, we shall ahow that the ptecedmgequationanBes fromthe elimination of a, betweenthé &Uow-
ingequationand ita differentialswith respectto a and j8:
RULEDMJ&FACE8. 845
.And by compartaonwith the preceding equationt we have
~+~=(t+M'')~'(a+m~8). If then we caU a+m.8, y the
problem ia redaced to eliminatey between the equatMM
RULED BUBFACES.*
410. On account of thé importanceof ruled surfaces,weadd somefurther AetaUaaa to this family of surfaces.
The tangent plane at any point on a genemtor evidentlycontainsthat generator, whichMone of the inflexionaltangenta(Art. 284) at that point. Each different point on thé gene-rator bas a différent tangent plane (Art. 107) which may beconetmctedas Mlows: We know that through a given point
Thetheoremsin thiasectionareptincipfdlytaken fromM.ChMÎM'aMemoir,(taetetet'9Cbt~Mpc~HM,t. xi., p. M, and &omMr.Cayley'apaper,ChmM<~and D~Mw3&M<M«<Ma<<K'Mtw~,VoLvtT.,p. t7t.
RULED8UMACES.846
can be drawna Urneinteraectingtwo given Unes namety, the
intersectionof the planesjoining the given point to the givenlines. Nowconsiderthree consecn~vegenerators,and throughany point Aon one~drawa line meeting the other two. Thia
line, paastng through du'ee cousecutivepoints on the ecTf&ce,will be the second m&ex!<MMltangent at .4, &Ndthereforethé
plane of thia line and the generator at Mthe t&ngentplaneat ~t la this constructionit is aupposedthat two coMecutive
generators do not intenect, which ordinarilythey will not do.
There may be on the mr&ce, however, Btngalar genemtONwMch are intersected by a coneeca~vegoaemtor, and in tMscase the plane containingthe two consecutivegenerators is a
tangent plane at every point on the generator. In apecialcasea also two conseent!vegenerators may coincide,in which
case thé generator !aa double Une on thé surface.
411. The anharmonicraio of four <an~M~~!aM<~MMM!~
througha ~KM'<tf<M*Me~Ma!<0that of theirfour points of con-
<tM<.Let three tixed lines J9, C be intersected by four
transveraassin points <M'<t"<?'&"& cc'e"c' Then the an-harmonie ratio ~&'&"&} ~cc'o"o"'))since either tneasare: the
ratio of thé four planes drawn through A and the four trans-
férais. In like manner {cc'c"<}= {<M'a"o'"}either measuringthe ratio of thé four planes through B (aeeArt. 112). Nowlet the three fixed Unesbe three consécutivegeneratorsof thé
ruled surface, then by the last ar~cle, thé transversals meet
any of these generatoraA in four points, the tangent planesat which are thé planes containing ji and thé tranaverBals.
And by this aritcle it bas been proved that the anharmonic
ratio of the four planes is equal to that of the points wherethe transversals meet A.
412. Ghren any generator of a raled Mu'face,we eau de-
Bcnbea hyperboloidof one sbeet, which ahaUbave this gene-rator in commonwith the ruled surface, and which shall alsobave the same tangent plane with that surfaceat every pointof their common generator. For it is evident&om the con-
struction of Art. 410 that thé tangent plane at every po!nt
347RULED SURFACES.
on a generator is fixed,when the two next consecutivegene-Kttom are given, and conMqaentlythat if two nded N))'<Me9have three consécutivegeneratomin common, they will touchall along the nrst of these generatora. Now any three non-
intersectmg right lines déterminea hyperboloidof ono aheet
(Art. ?6) the hyperboloidthen determinedby any generatorand thé two next consecutivew!Htouch the given ear&ce as
required.In order ta aee thé full bearing of the theoremhère enon-
ciated, let us suppose that the axis of lies altogether in anysurfaceof the M'"degreo, then every term m ite eqaat!ommustcontain either a) or y; and that equation arranged accordingto the powera of a) and y will be of the form
where < e~ denote fnnctiona of <: ofthe (<t-1)" degree, &c.Then (see Art. 107) the tangent plane at any point on the axis
will be M'a!+ e~'=0, where M' denotes thé remit of snb-
stttuting in M~, the co-ordinates of that pomt. ConveKety, itfollows that any plane y=<<M: touches the surface in M-l 1
points, which are determined by the equation M~+~=0.If however «, have a common &ctor «~ M that theterms of the nrst degree in a! and y may be written
~(K,+e~)=0, then the equation of the tangent planewill be M'a;+e'=0, and evidently in this case anyplane y=Ma: will touch the surface only ïn n -p -1 pointa.It is easy to see that thé points on thé axis for which u, = 0arc double points on the surface. Now what is aMerted in the
theorem of this article is, that whcn the axis of is not an
isolated right line on a surface, but one of a System of rightlines by which the surface is generated, thcn the form of the
equation will be
so that the tangent plane at any point on the axis will be the
same as that of thé hyperboloid t<.B+ ey, viz. M'.B+ c'y= 0. And
any planey =M.B will touch thé surface in but one point. Tho
factor indicates that there are on each generator a-2 2
points which are double points on the surface.
RULED SURFACES.848
413. Wo can vorify the theorem juat stated, for an !m-
portant daM of ruled aur&cea, v!z., thoee any generator of
which cm be expreMedby two equationsof the form
where a, a', b, &o.are linear fonctionsof the oo-ordinates~and t a variable parameter. Then the equation of the surface
obtainedby eliminating t betweenthe équationsof the gene-rator (Bi~Aef.w&Mt, p. 84), may be written in the form of
a determinant, the first row and first colamn of wh!chare
identical, viz., (o&'),(<M'),(ad*), &c. Now the line <M'M a
genemtor, namely, that answering to (=00; ami we have
jost proved that either a or o' will appear in every term bothof the &9t row and of the fmt colnmn. S!nce then everyterm in thé expanded determinant contains a factor from the
Ëmt row and a factor from the first column, the expandeddeterminant will be a funct!on of, at least, thé seconddegreein <tand a', exeept that part of it whiehia multipliedby (<!&'),the term commonto the 6rst row and Srat coinmm. But that
part of the equation which is only of the first degreo in a
and a' determines the tangent at any point of <M' thé raledsurface is therefore touched along that generator by thé hy-
perboloid«&&a'=0.If ot and b (or a' and ~') represent the same plane, then
the generator N<t'intersectathe next consecutive,and thé planea touches along its whole length. If we had &=:~ft,&=~the terms of the nrst degree in a and a*would vanlab,and<M'would be a doubleline on the enrfaoe.
414. Betaming to the theory of mled swr&cesin general,it !a evident that any plane through a generator meets the
surface,in that generator and in a cnrve of the (n 1)'" degreemeeting the generator in K–Ï points. Each of thèse points
being a double point in the carre of section is (Art. 288) ina certainaenaea pointof contact of the plane with the surface.But we have seen (Art. 4M) that only one of them ia properlya pointof contactof the plane; the other a 2 are nxed pointson the generator, not varying as the plane throngh it is
ttUMH6UBFACE8. 349
changed. They are the pointa where t'hia generator meetaother non-<!OMecatîvegenerators, and are pomts of a doublecurve on the surface. Thus then o <Jtetoruled aurfaceM)general~M a doublecurve which te met by every ~eMr<!<<M'M M–22
~oM<.b.It may of eouraehappen that two or more of thèseM-8 2pointsmay coincide,and that the multiple curve on thesurface may be of higher orderthan the second. In the caseconsideredin the last art!cte it can he proved (see Appendixon the Order of Systemsof Equations)that the multiple cnrve
A ruled surface having a double line will in general nothave any cuspidal line unless the surface be a developable,and thé section by any planewill therefore be a curve havingdouble points but not cusps.
41&. Consider now thé cône whoso vertex is any point,and which envelopes the surface. Suice every plane througha generator touches thé surface in some point, the tangent
planes to the cone are the planes joining the series of gene-ratoïs to the vertex of the cone. Thé cone wul, in general,not have any stationary tangent planes: for such a plane would.arise when two consécutivegenoratomlie in the same plane
passing through the vertex of the cone. But it is only in
spécial casesthat a generator will be interaectedby one con-
sécutive the number of planes through two consécutivegene-ratom is therefore finite and hence one will, in general, not
pass through an assumedpoint. The dasa of the cone, beingequal to the number of tangent planes which can be drawn
through any Une through thé vertex, !s equal to the numberof generatora which can meet that Ime, that îs to say, to the
degree of thé surface (seenote, p. 13~). We have proved now
that the c~M of the cône Mequal to the degreeof a sectionof the sorface and that the former bas no stationarytangent
planesas the latter bas no stationary,or cuspidal,points. The
equations then which conscct any three of thé singalandea
RULRD SOM'ACES.850
of a curve prove that the number of double tangent planesto the cone must be equal to the number of double pointsof a section of the surface; or in other worda, that the number
of planes containmg two generatora which ean be drawa
through an assumed point, is equal to the number of pointaof intersection of two generators which lie in an assumed
pitme.~
416. We ahall !Uastr&te the preceding theo~jr by an enu-
meration of some of the singularities of the ruled surface gene-rated by a line meeting three fixed directing cnrvea, the degreesof which are m,, ~.f j'
The degree of the surface generated ia equal to the number
of generators which meet an aMumed right iine it is there-
fbre equal to thé number of intersections of the curve m, with
the raled surface having for directing curves the curves M,, m.aad the assumed line; that is to say, it Is m, times the degreeof thé latter surface. The degree of this again la, in like
manner, times the degree of the ruled surface whoae directingeurves are two right Unes and thé carre M,, while hy a repe-tition of the same argument, the degree of this last is 2m,.It follows that the degree of the ruled surface when the
generators are curves ?! <K,,<M,,M2wt~The three directing curvea are multiple lines on the sar~ce,
whoae ordera are reapectively ~M,, M,<M,,M~M,. For through
any point on the firat curve pao mm, generatom, the inter-
eeotiomenamely of the cones having thia point for a common
vertex, and Miting on the eurvea <M~M,.
417. The order of the ruied surface being 2m~,m~ it
foUows, from Art. 414, that any generator M mteraected by
2M,m~,w,–2 other generators. But we have seen that at
the points where it meets the directing curves, it meets
(o~m,–l)-t(ot,<!t,-l)-t(M)~–l) other generators. Come-
Thèse theoKmt are Mr. Cayte/t. ChmM~ and D«M)t ~&<tt-
<xa<t:M<JTnt~<t<,Va!, inn., p. 171.t 1 publlshed a dieouasion of tM< tfm&ce, OMHM~and ~)«MM
Jtta(~<HM«M!Jot<)w«!,VoL ~nï., p. 46.
RULED SURFACES. 351
quently it must meet 2M~)H,-(m,M~+m,M,+M,Mj+l gène-
rature in pointa &oton thé dtrecttng curvea. We ehaU estabUahthis reault independently by seeking the number of generatonwhieh can meet a given generator. Let us commence bydetermining thé degree of the ruled surface whose directingcurves are the curves m~ m,, and the given gcnorator, whieh
is a line resting on both. In the Ërat place this right line
is a multiple line of the order M,t, since obvionety,
tbrough any point of It can be drawn thia number of
lines (distinct from the given iine itself) meeting the curves
<K~ m,. But the section of the surface by a plane throughthe given Une, will be that Une !tBetf (M,w,-1) times, togetherwith the (0~–1) (o~–l) generatom, obtained by joinmg anyof the points where the plane meeta the curve to one of
those where it meets the curve <?“. Thus then the degreeof thé section (and therefore of the surface) is
MuMplymgthis nnmberby ?“, we get the number of pointswhere this new rnled eur&ceis met by the corve M,. But
amongst theae will be reckoned (tM~-l) times the pointwhere the given generatormeetsthe curve M,. Subtractingthis
nomber then, there remain 2m~Mj,-M!,M,M,-M~,+l 1
points of the curve throngh which can be dt'Mra &line
to meet the curves M,,M,, and the assumed generator. But
this is in other words the thing to be proved.
418. The ruled sm'&cewill contain &certain number of
doublegeneratora,thosenamelywhichmeet one of the directingcorves twice and thé other two once. The number of Mch
lines resting twice on the ourve m, la proved by reMonmgsunilar to that used before, to be <M~ t!mes the degree ofthe ruled surface generatedby a nght HneKst!ng twice on
m, and a!so on an arbitrary line. Now if h, be the numberof apparent double points of the carve w,, that ia to say, thenumber of lines whieh can be drawn through an assumed
point to meet that curve twice, it is evident that the aMamed
r!ght HBewill on tbis ruled surfacebe a multiple line of thé
MtE& aOBFACES.362
order~t, amdthe sectionofthé ruled surfaceby a p!mMthroughthat !iiM,will be that lineA,times togetherwith the (~ -1)lineajoining any pair of the points where the plane cuts the
curve M~ Thé degree of this ruied surface will then he
~+~m,(N~–l), and thé total number of double generatomon thé original m!ed aar&ce is
Il
1 am unable to give the order of the double cnrve in general,bat in the particular case where one of the directing auveeis a right Une,and the other two curvesof the degreem,,m,,it ia evidentthat the sectionby any plane through thé directingright limeconsists of that right Une M,m,t!mea togetherwith
~M, lines mtemectiNgin ~m~,(~ i) (~ 1) pointanot onthé directing eurvea This latter therefore would appear tobe in this case the order of the nodal curve, «M&Mit intersect
the directing line in a certam number of points,which, if so,must be added to the order of the carre. There are, of ooame,hesîdes,double generators,as determined in the firat part ofthia article.
It is easy to see, in like manner, that the surfacegeneratedby a right line resting twiceon a curve mand on a right line,will hâve, besides ita doublegenerators, a doublecurve,whoae
order is, a< &<M<,~K(m-l)(M-2)(<a-3).
419. The degree of the mied surface, as calculatedbyArt. 416, will admit of reduction if any pair of the directingeurves have points in common. Thas if thé corvea mghave a point in common,it is evident that the cone whosevertex is this point, and base the cnrve <M,will be inelndedin the system, and that the prder of the ruled surfaceproperwillbe reduced by M,. And generally if the three pairsmadeont of the three direetimgcurves have common respectivelya, ~9,'y points, the order of the raled surfacewill be reduced
by ~a+ Mt,/3+ M,'y.* Thas if the directing linesbe tworight
Myattentionwasoa!)edbyMr.Cayleyto<MtreductionwMchtakMplacewhenthe~ireetingMrvMhavepointaineommon.
BULEDSURFACES. 353
A A
!meaand a twistedcubic,tho surface is in gênera! of the sixth
order, but if oach of tho lines intersect t!t0 cubic tho ordcr h
only of tho fourth. If oach interacet it twico tho surface i.,
a quadric. If one intersect it twico and thé other once, thesurface is a skew surfaceof the third dogree on~which theformer line is a doublelino.
Again, let thé dircctmg curvesb&any three plane section!!of a hyperboloidof one shect. Accordingto tho geneml theorythc surfaceought to be of tho sixteenth order, and let us sechow a reduction takes place. Each pair of dirocting cnrvcshave two points common; namely, tho points in which thoUneof intersectionof their planes meets tho aar&ce. And tho
complex surface of thé sixteenth order consistaof six conesofthe second order, togother with tho original quadrMrockonedtwice. That it must be rcckoncdtwico, appears from thc factthat tho fourgencrators wMchcan be drawn through &nypointon one of the directingeurves, are two Imes bolongingto thc
cones,and twogeneratorsof thé givenhyporboloid.In general, If wo take as diMctmgcurves thrce plane sec-
tions of any ruled surface, thé equation of tho rnled surface
generatedwill hâve, in additionto tho conesand to thé original
sarfaco, a factor denoting another rnicd surface which passes
throngh thé given curves. For it will generally be possibleto draw lines, meeting all threo curves, which a~o not gene-rators of thé original surface,
420. Retnrning to thé CMCof raled surfaces in general;weknow that a eonesof planes throngh any lino and a soriesat right angles to them form a systemin invointion, thé an-harmonie ratio of any four being equal to that of their'four
conjugales. It followsthon, from Art. 411, that the systemformedby the points of contactof any plane, and of a pianoat right angles to it, form a systemin involation or, in other
words,thé systemof pointswhereplanesthrough any genaratortouch the surface, and where thoy are normal to the surface,form a system in involution. Thé centre of the aystcm is thé
point where thé plane which touches the surface at innnity,is normal to the surface and by thc known properties of in-
RULED SURFACES.864
volution, the distances from this point of the pointa where
any other plane touches and is normal, form a constantrect-
angle.
421. ?%<normalsto any ruled MO~Mealong<tM~~e)Mf<t<M',
generatea Ay~~Mtc paraboloid. It is evident that they are
all parallel to the same plane, namely thé plane perpendicularto the generator. We may speak of the anhajmonic ratioof four linea parallel to thé same plane, meaning thereby thatof four parallela to them through any point. Now in this
sense thé anharmonic ratio of four connais is equal to thatof thé four correspondingtangent planes, which (Art. 411) is
equal to that of their points of contact,which again (Art.419)M equal to that of thé points where the normals meet the
generator. But a system of linea parallel to a given planeand meeting a given Une gener&teaa hyperholicparaboloid,if the anharmonic ratio of any fonr la equal to that of thefourpoints where they meet the line. This propositionfollows
immediatelyfrom its converse,whichwe can easilyestablish.The points wherefour generators of a hyperbolicparaboloid
intersect a generator of the oppositekind, are the points ofcontactof the fourtangent planeswhichcontainthese generators,and therefore the anharmonicratio of the four points is equalto that of the four planes. But thé latter ratio is measured
by thé four lines in whieh these planes are intersectedby a
plane parallel to the four generators, and these intersectionsare lines parallel to these generators.
422. Thé central points of the involution (Art. 419) are,it is easy to see, the points where each generator is nearestthe next consécutive,that is to say, the point where each
generator ïs intersected by the ahortest distancehetween itand its next consecutive. The locus of the points on the
generatora of a ruled surface, where each is closest to thénext consecntive,is called thé ?MM striction of the surface.It may be remarked, in order to correct a not unnattu'almistake(see ZacroM!,Vol. in., p. 668), that the shortestdistancebetween two consécutivegenerators is Mo<an elementof the
BULEOSURFACES. 355
AA2
line of etrietion. In fact if ~<t,J9&,Ce be three consecutive
generators, a&the shortest distance betwecn thé two former,then &'<?the shortest distance between the secondand thirdwill in general meet Bb in a point &' distinct from &,andthe element of the une of striction will be at' and not ab.
Ex.1. To6ndthelinoofstrictionof thehyperbolicparaboloid
Any pair of generatoramay be expKMedby the equ&tioM
Both being parallel to theplane
their shortett distance is pM-
pendteola)' to thia plane, and therefore lies m the plane
«"-fiwhiehhtemeetsthé &st genemtotIn thépoint < c -gO+U-AP.'When thé two genetatoK appMach to toinddeMe, ~fe h&T6foz theeo-ordin&tM«f thé point where either M inteKMted by thNr ahottMtdMianee
The lino of tt~ction ia therefore the parabola in whieh thia plane euuthe Bnt&ce. The same surface considered M generated by the lines
of the other system has another Une of striction lying in the plane
( 356)
CHAPTERXIII.
SURFACESDEBIVEDPBOMQUADRICS.
THE WAVE SUBFACE.
423. BEFOBEproceedingto surfacesof tho third degrec,we think it more simple to treat of surfaces derived from
quadrics, thé theory of which is moro doscty connectcdwiththat explained m preceding chapters. Tho equation of tho
surfaceof centres bas been already givon (Art.208), and we
proceed now to define, and form the equation o~ Fresnel'sWaYe Sur&ce.*
If a pcrpendicalar through tho contre be ereoted to tho
planeof any central sectionof a quadrie, and on it tengthsbotaken equal to the axes of thé section,tho locusof their ex-tremities will he a surfaceof two shects which is caUedthéwave surface. Its equation is at once derived from Arts. 97,98, whero thé lcngths of thé axes of any sectionare expresocdiu terms of tho angles whicha perpendicnlarto its planemakeswith tho axes of thé surface. Tho same equation thon ex-
presses the relation which tho length of a radius vector to thewavo surface hears to thé angles which it makes with thoaxes. The equation of thé W&YeSurface is therofore
See Preimet, J)~M<'A'«de l' Inttitut. Vol, TH.,p. ]SO,pnUMted 18!
ntE WAYESUEFACE. 357
From thé 6mt form it appcars at once that the intersecttonof thc wavo smf&ccby a conccntnc sphere, is sphero-comc.
424. The section by one of thé principal planea (e. the
plano a) breaks up into a cMe and e!I!pM
This is a!ao goometrically codent, since if we consider anysection of thé gencrating quadric, throngh tho axis of oneof tho axes of that section is equal to c, whi!othe other axislies in thé plane a;y. If then wc crcct a porpendicular tothe plane of section, and on it take portions eqnal to cachof those axes, tho extremitica of one portion will trace out acircle whose radius is c, whHotho locus of the extremities ofthé other portion, will plainly be the principal section of the
generating quadric, only tnrned round throngh 90".<'In cachof the principal planes tho sur&ce bas 6)ur, jou~ pointsi
namely, tho intersection of the cMo and ellipsejust men-tioned. If a; y' bo the co-ordmatcsof~ne of thcse intersec-
tions, the tangent cono (Art. 239), at this double point, basfor its equation r-
Tho gcncratmg quadric boing supposa to bé an ellipsoid, itis cvtdcnt that in thé case of the section by the plane tho
circle whose radius is c lies alto,getherwithin tho ellipso whoseaxes arc a, b and in tho CMCof thé section by the ptane <e,thc circle whoso radius !a a, lies&ttogcthorwithout the ellipsewhoso axes are 6, c. Real double points oceur only in tho
section by tho plane they aro evidently the points corrc-
sponding to thé circular sectionsof tho generating ellipsoid.The section by the plane at infinity also breaks up into
factors a~+~+< <t*a:'+&y+c'< and may therefore atso boconsiderodas an imaginary circle and ellipse, which in like
mannergive rise to four imaginary double pointsof tho SHr~coSttnatcdat infinity. Thus the surfacehas iu all sixtcen nodal
pointa,only four of winch arc real
THEWAVESURFACE.3S8
42S. Thé wave surface is one of a dasa of surfaceswhich
may be calledapsidal «o~tcew. Any surface being given, ifwe assume any point as pole, draw any section through that
pole, aud on the perpendicular through the pole to the ptaneof section, take lengths equal to thé apM<M(that ia to say,to the maximumor mmimam) radît of that section; then thelocus of the extremities of these perpendiculars19the apaidalsurface derived from the given one. Thé equation of the
apsidal surfacemay alwaysbe calculated,as in Art. 98. F!rst
form the equation of the cône whose vertex is the pôle, andwhich passeathrough thé intersection with the given surfaceof a aphere of radius r. Each edge of this cone is proved
(aa at Art. 98) to he an apsidal radius of the sectionof the
surface by thé tangent plane to the cône. If then we form
the equation of the reciprocal cone, whoseedgea are perpen-dicuiM*to the tangent planes to the first cone, we ahall obtainall the points on the apsidal surfacewhich correspondto the
tangent planes of the assumed cône. And by oonsideringr
variable, in the equation of this latter cone, we have the
equation of the apsidal aurface.
426. If OQ be any radias vector to the generatingsar&ce,a&dOP the perpendicularto the
tangent plane ttt the point Q, then
OQ will be an apaidal radius of
the section passing throngh 0~and through OB which is sop-
posed to be perpendicularto thé
plane of the paper POQ. For
the tangent plane at Q passes
through PQ and is perpendicnlar to thé plane of the paper;thé tangent l!noto thé section QOR lies in the tangent planeand is therefore also pefpendicntttrto thé plane of the paper.Since then OQ is perpendicular to the tangent line in thesection ~OJ!, it is an apaidal radins of that section.
It followsthat 0?~ the radius of the apaidalsurfacecotTe-
apoBdiBgto the point Q, lies in the plane POQ and is per-
pendicularand eqnal to 0~.
THE WAVE SURFACE. 3ô9
427. ne ~et~MoA'ott&n'<A<tan,gentplane ta <Aeapsidal<t<<~Meat r a&oin theplane JP<?~ <Nt~M~tp~&MJiafand egtKt!to OF.~
Consider Srat a radius 02" of the apsidal surface, :nde-
finitelynear to 02~ and lying in the plane TOR, perpendicularto the plane of the paper. Now OT 19by definitionequalto an apaidal radius of the section of the original eur&eobya planeperpendicularto 0?*, and this planemnst pass throughOQ. Again an apsidal radms of a section is equat to thenext consecutiveradius. The apaidal radius therefore of asection passing through OQ, and indefinitely near the planeCM, wîU be equal to OQ. It follows then that <3y=<?T*,and therefore that thé tangent at T to the section TOR ia
perpendicularto OT, and therefore perpendicular to the planeof the paper. Thé perpendicular to the tangent plane at T
muet therefore lie in the plane of the paper, but this is theBmtpart of the theoremwhich was to be proved.
Secondiy, oona!deran indefiuitely near radius 02*' in the
plane of the paper; this will be eqnat to an apsidal radiusof the section JÏOQ', where Oq M mdenmtely near to OQ.
Bat, M befbre, this apsidal radius boing Indennîtely near to
Oq will be equal to it, and therefore 02~" will bo equalas well as perpendicularto t~ Thé angle then fTO Is
equal to ~~0, and therefore the perpendieular 0~' !s eqnaland perpendicularto OJ~
It Mowa &om the symmetry of the constmction that îfa surfaceJ[ iathé apsidalof B, then converselyB is the apsidalof A.
428. Thepolar reciprocalof au apSMMsurface,<p!'<Arespectto theorigin 0, is thesameas the apsidal of thefectproc~, <c!7A
respectto 0, ~<~ ~tpeKsurface.For if we take on OP, OQ portions !nver6etyproportional
to them, we shall have C~, Oq, a radius voctor and corre-
spondmgperpendicularon tangent plane of thé reciprocal of
Theaetheoremearedue to Prof.MacCuUagh,K-<M«!<'<)'<MMo/' tAo
RoyalIrish~[e«ef<my,Vol.XVI.
TIIE WAVE SCBFACE.360
tho given surface. And if we take portions equal to thcscon thé lines OtS,OT whieli lie in thc!r plane, and are rcspoc-tively perpendicular to them, thcn by the !ast article wc sbaUbave a radius vector, and correspondingpcrpcndieularon tan-
gent piane, of thé apsidalof thé rociprocal. But these longths
being mvcrsely a&OS, OT are also a radius vector, and per-pendicular on tangent plane of thé rcciprocal of the apaidal.T!M apsidal of tho reciprocal is therofore the samo as tho
rcciprocal of thc aps!dal.In particular, thé reciprocalof tho wavo surfacegcneratcd
from any euipaold,is tho waïc surfacegenoratcd fromtho rcci-
procalellipsoid.We might have othcrwtso seen that tho reciprocal of a
wavo surface !a a Eur&cûalso of the fourth degrce, for tho
rcciprocal of a surface of the fourth degree is m general oftho tMrty-mxthdcgrec (Art. MO) but it is proved,as for planecurvea,that each doublepoint on a surface redaces the degrecof ita rcciprocalby two; and wc have proved (A]t. 424) that
the wavcsurfacebas s!xtcendoublepoints.To a nodal point on any surface (whichis a point through
whieh can bo drawn an infinity of tangent planes, touchinga conoof the seconddcgroe) answorson the reciprocalsurfacea tangent plane, ha.vmgan infinity of pointa of contact, lyingin a conic. From knowing then that a wavo surfacebas fourreal double points, and that the reciprocalof a wave surfaceis a wave surface, we infer that tho wavo surface bas four
tangent planes which touch ail along a conic. We shall nowshow goometncauy that this conicia a orde.~
429. It is convenientto premiscthe followingiemmas:
LEMMAI. If two lines passingthongh a 6xcd point, andat right angles to each othor, movc each in a fixed plane, tho
Sir V. R. HamiltonSKt showedthat the wavesurfacehMfournodcs,the tangentplanesat whichenvelopecones,and thatit hasfour
tangentplaneswhichtouchalongeircles,J~'aoMt<<oM<o/'<~ Royal~<*MA~ca<~N<y,Vol.XTtI.,p. t32. Dr.LloydexperimentallyverinMlthéopticalthcorcnn.thBnccderived,fM~,p. tM. T)ic gt.ùmetttealmMttig:Hlonswhtchfollowarcducto Profc~).'.\fac Odiagh,p.2M.
T!tE WAVESURFACE. 361
piano contaiBmg thc two lines ccvelopca a coao whMO sectiona
parallel to tho fixed phmca arc parabolas." The ptauo of the
paper is supposcd to bo parallel to ono of thé &Md planes,and the other fixed plane is aappOBed to pass titrough the
!mc JM~. Tho Sxed point 0 m whieh thé two Unes mtersect
is supposed to be above thé paper, P being tho foot of the
perpondicularfrom tt on tho planeof tho paper. Now let OJ? be one
positionof the lino which movcsin
tho plane OJM~then thé other linoOA whieh Is parallel to the planeof tho papor being perpendicularto
Û.Bsnd to OP is perpendicularto
tho plane O.NR But tho planeC~tZ<interaects tho plane of tho
paper ina linojBÏ*parallol to O~t,and thereibroperpendicularto JXR And thé envclopeof BT ie ovidentlya parabola ofwhich P la thc &)cnsand 3fy thé tangent at thé vertex.
LEMKAII. "If a lino OC bc drawn pcrpcndicular to
Û~J?, it will goncmte a cone whoso circular sections are
parallel to the fixed pianos." (Ex. 4, p. 86). It is proved, asat p. 106, that tho locus of C is the polar rectproca!,with
respect to 2~ of the envclopc of 2?~ The locus is thcre&rca cirelopassing throngh H
LEMMAIII. If a central radins of a quadnc moves in afixed plane, the corrospondingperpcndicuIaron tangent planealsomovesm a nxcd plane." Namely, tho plancperpcndtculat'to tho diamoter conjugato to tho tiret ptanc, to which tho
tangent plane must bo parallel.
430. Suppose now (soc ngnre, Art. 426) that the plane0<?~ (where OR is perpondticularto the pianoof tho paper)is a circular sectionof a quadric, then OT !s tho nodal rad!usof tho wavo sNr&co,which remains thé aamowh!lc OQ movesin thé plane of tho ctrcular sections; aud wc wish to findthc conogcneratcd by Hut OtS'is pm'pcmitcularto OJSwhich tuovcs in thc ))t)mcof tho ch'<u!.u'sccttOiMand to ~7'
THE WAVE SURFACE.362
wMch movesin a Sxed plane by Lemma III., therefore OS
générâtes a cône whoM circular seotionaare parallel to the
planes JPOJB,~M. Now T is a fixea point, and TS !a
parallel to the plane POR, thereforethe locaa of the pointM a circle.
The tangent côneat thé node is evidentlythe reciprocalofthe cone generated by 0~ and is therefore a cône whosesectîons parallel to the aame planes are parabolas.
Secondiy, supposethe line OP to be of constant length,which will happenwhen the plane POJï Ma ttansveNesectionof one of thé two right cylmdera whieh e!rcumMrlbethe
ellipsoid, then the point 8 ts 6xed, and it !s provedprectaelyas m the first part of thia article that the locus of T is acircle.
431. The equationa of p. 178 give immediatelyanotherform of the equation of the wave surface. It is evident
thence, that if 8''be the angles which any radins vectormakes with the Unes to the nodes, then the lengths of the
radius vector are, for one sheet,
It follows hence also that the intersectionsof a waye atuAcewith a series of concentric sphères, are a series of confocal
sphero-conics, For m the preceding equations if p or p' be
constant, we bave ~±6' constant.
432. The equation of the wave surfacehas also been ex-
pressed as followsby Mr. W. Roberts in ellipticco-ordmatea.The form of the equation
368THE WAVE SURFACE.
ohove that thé equation may be got by ottmmatutg between
thé equa~oM
Ghnng any sénés of constant values, the firat equationdenotes a series of eonfocalquadrica, the axis of <!being the
primary ax!e, and the axis of x the kMt. Smce !a alwaysleœ than <t*and greater than c*, tho equationalways denotesa hyperboloid,which will be of one or of two sheetsaccordingaa Mgreater or lésa than b'. The intersectionsof the hyper-boloideof one sheet with corresponding spheresgenerate oneaheetof the wavesurface,and those of twosheetsthe other.
Now if the sur&ce denote a hyperboloidof one sheet, anjif ~t, v denote tho primary axes of three conjbcal surfacesof the systemnow under considerationwhichpass through any
point, then the equationgives us f" <~ bnt (Art. 169)
The general equation of the wave mr&ce atso implies
~'+~= «*+&< but this denotes an imaginary locus.
Since, if is constant, p is constant for ono sheet and f
for the other, it follows that if through any point on the sur-
jËMebe drawn an ellipsoid of the same system, it will meet
one eheet in a Une of curvature of one system, and the other
sheet in a Une of the other aystem.If thé equations of two snr&ceB expressed in terms of
X, v, when diiferentiated give
THEWAVE8URFACË.SCt
f =constantc&tsat right angles any whose equation Mof thefonn <~(\t)=i0. The hyperboloid thercfore, f!=constant,cats at right angles one sheet of the wave surface,while itmeets the other in a line of cnrvature on thé hyperboloid.
483. The planeof any )'<!<?«veetorof the wavesurfaceandthe c<Mva!po~!H~~e<pM<?MM&tfonthe tangentplane, makesegtM?angles with the planes Mt~KM<XC<'<M'and the nodallines. For thé ërst plane is perpendicalarto OB (Art. 426)wbieh !aan axis of tho section ~OJB of the gcnerating etHpsotd,ana tho other two planes are perpcndicnlar to thé radu ofthat section Tthoaclengths are 6,thé mean axis of the ellipsoid,and thèse two equal Unes make equal angles with the axis.Tho planesare evidently at right angles to each other, which
are drawn tbrough any radius vcctor, and the perpendicalaraon thé tangent planes at the points whero it meets thé twoshects of thé surface.
Reclprocating thé theorem of this article we see that thé
plane tbrough any line throngh tho centre and through oncof the points where planes perpendicular to that lino touchthe sar&ee,makes equal angles with the planes through thesame lino and through pcrpcnd!calarsfrom thc centre on the
planes of circular contact (Art. 430).
434. If tho co-ordmates of any point on tho generatingelHpsoidbe a!y. and the primary axes of confocalathronghthat point a', d'; then thé squares of tho axes of tho section
parallel to the tangent plane are «*-o", <a" which woshaU caH p*, p". Thèse then givo the two values of the
radius vector of tho vavo surface, whosodirootion-cosinesarc
~-) Wc shall now calcdate the leugth and thé
dircction-cosinesof the perpendieular on thé tangent plane at
cither of thépoints whero this radins vectormeets thé surface.It was proved (Art. 427) that thé required perpendicular is
equal and porpendicalar to the perpendicularat the point whero
the cllipsoidis met by ouo of thé axes of the section and
the dli'cctton-cosincsof this axis arc Tho
TUE WAVE RUBFAPE. 365
co-ordin&teaof ita cxtromity arc then thcao several coa!nc!
multiplicdby p, and tho d!rcctMn-cos!ne)!of the correspondingperpendicularof the ellipsoidare
Th!a then gives tho length of tho perpend!cular on tho
tangent plane at tho point on tho wave surface which we arc
conmdenng. Ita dtTeetton-cosmcsare obtained from the con-
sideration th&t it is perpendicalar to the two linea whose
direction-cosmeB_~rerespeetively
Fonmmg by Art. 15 tho directton-cosmea of a Hae pcrpondicularto these two, wc find, after a fow rednctïon~
In fact it is verifiod withont di&cnlty that tho lino whose
direction-cosineahave boen just wnttcn !s perpendicular tu
tho two preceding.It follows hence also, that thé equation of the tangent
plane at the same point M
In uko manncr thc tangent plane at tho other point whcrû
the same radius voctor mceta thé atu'&cois
THE WAVE SCRPACE.S66
435. If be the angle which the perpendicnlaron thé
tangent planemakes with the radiusvector,wehaveP= p cos~
but we have in thé !&Btarticle proved JP'ni Hence')! F'P
co9*~==- tan*~=~. Th!s expreaBionmay he trfUM~
formed by meaaBof the vaines given for p and jp' (Art. 178).We have therefore
In this fMin thé expresstom!s analogonato thé valuefor theanglebetweenthénormalandcentrâtradmavectorof a planee!Hpse,vîz.,
In the case ofthe wave sai&ce it is manifet that tanvanifthea
only when p'=<~ or c, and becomesindeterminate when
~=~=~
436. The expresnon tanCe=~ leada to a construction for
the perpendicahra on the tangent planes at the points where& given radius vector meets the two sheets of the surface.The perpeNdtculcuramust lie in one or other of two hed
planes (Arts.433, 434), and if a plane be drawn perpendicularto the radius veetor at a distancep, it is evident from the
expresmonfor tan~, that is the distanceto the radius vectorfrom the point where the perpendicularon the tangent planemeeta tbis plane. Thus we hâve the construction,"Draw a
tangent plane to the generating eïBpsotdperpendicularto thé
given radius vector, from Its point of contactlet &I1perpen-diculara on the two planea of Art. 438, then the lines joiningto the centre the feet of thèse perpendiculara,are the perpen-dîcnlam required.
THEWAVESUBFACE. M7
We obtain by reciprocationa MmUarconstruction,to de-temine the pointawhere planes parallel to a given one touchthé two sheets of the surface.
437. 1 have sometimesfound it convenientto traM&nnthe equation of the surface, as at Art. 180, M as to makethe radins vector to any point on thé surfacethe axie of <and the axes of the correspondingsection of the generatingeH!pso!dthe axes of x and y. We may write thé equationof the surfacem thé form
Now .~+y*+&' remains unaltered by transformation, and we
have given, Arts. 183,184, the transjbrmadoM of
JL~1 1 1
~+~+~1
and have also uaed the identical equation
~+oV+a'&'=~+~+F"p''+~'+~
THEWAVESURfACE.3C8
By expanding thé tcrms of the Bcconddegrcc, the valuesof thé pnnctpal racHiof cnrv&tnro~and tbe directionsof cur-vature can bc establiahed,but 1 have arrived at M ro!ult9of
importance.
438. Tho equation of the rociprocalof the w&YOsurfaceX*
M got by writing for a, &c., in the equationof the warca
surface; and if this be traasfbrmedaa in the precedingarticle,it becomes
We know that thé Bnrf&ceis toachcdby the plane a'=–, 1Pand if we put in thia vaine for s, va <md,M wc onght, a
carye havmg for a douMopoint tho point y==0, a!c=~
If in tho equationof the carvo ~cmakey==0, wogetPP
from whichwc learn that that chord of the outer sheot of thowava surfacewhieh joins any point on the mnor shect to thefootof tho perpencUcularfromthe centre on the tangent planeM bisootcdat thc point on the inner shoot. The maoxioNal
tangcnta are paraUel to
a rcsdt of which 1 do not see any gcomctdctd Intcrpret&tton*
1 have no apMe for a diMUM!onwhat tho lines of curvatm'e on the
wave <ntfMeare tM<,though a hMty asaertion on this subject in Crcllo's
toutMt haa led to mtetetting investigations by M. Bertrand, C&mp<e<~M~«<,NoT. ÏM8t Combescun and Drioschi, Tortolini's Annali di ~<a<<t-
matica, VoL H., -pp. 13S, 278. It M worth while to cite an observathmof Brioschi, that if in the ptMMIl + my + M!!*'<~) m, n, be &nct!oM
SURFACESMNVBD FBOM QUADNM. 369
BB
489. We <M1next consider the autace jM~t!M to a givenquadric,that is to say, the surface which may either be definedas the envelopeof planes parallel to the tangent planes ofthe quatMo, eaA at a given distance irom them; or elae asthé locMofthe pointstaken on thé normaleat &Sxed distance
from the surface,(Bf~r <?&!?<CM~e~,p. 878). It Mevidentthat the apherewhosecentre is any point onthe paraUetsurface,and radius thé given distance)will touch the original qttadnc.We can then most easily form the equation of the parallelear&ce hy exprosMng(Att. 137) the conditionthat the given
w* z*qnadnc + + T* may be touchedby the aphere
This is done by forming the dMeriminantwith respect toof a MquadratiewhoeecoeSdenta are given p. US,but wHch
may be written in the form
Theresult representssurface ofthe tweMhdegree,but which,when we make &==0,tedoces to the quadrie taken twice, to-
gether with the imaginary developable (Art. 803) which en-
velopes all qattdrtcaconfbcal to the given one. This ret~Syappears fmmthe form in which the equation of the MqtMdrattchaaheenwnttetL
440. The locaa of the feet of perpendtcnlamlet Ml, from
any fixed point, on the tangent planes of a surface ia a de-
oftwovariables«,e,aainArt.3?3,thenthepluewiUenvelope<tmt&eein whiehcuMMof the &m!Ke9<t coMtant, eoaahmt,witt,et theirintetMedon,be touohedbyoonjugatetangentaof the Mt&ce,If theeon-dMonbe MNhtd, J
where the su&M<t, 2, denotedi&ten&nion wM)Mfpeet to u and eKq~Tdy!wMle~Mr~wUte<tt<tt)~htM(;te)!if
(<*M* +tt*) (<A+ attm,+M~) ° (%+mm~+<M))(%+ otM~ MM.).
SURFACES DERIVED FROM <)PAPJ!!CS.8?0
rived surface to which French tnathematicianshâve of late
thought it worth while to give a distmctivename, "podaire,"which we ahall translate M thé pedal of the given surface.From the pedalmay, in likemanner, be derived&newsurface,and from this another, &c. forming a seriesof second, third,&e.pedab. Again, the envelopeof planesdrawn perpendicularto the radâ vectores of a surface, at their extremMea ? a
snrfaceof whîch the given surface iathe pedal, and whtch we
mayca!l thé ËKt negative pedaL The surfacederivedin Ukemanner &om this !a the aecoad negative,and so on. Pedal
curves and aar&ceshâve been studied in particular by Mr. W.
Roberts, JMw~, Vois. X. and XK., by M. Tortolini, and byMr. Hirst, Tortol!m'a ~K< Vol. n., p. 95. We shall hère
give some of their resnits, but must omit the greater part of
them, which relate to problemaconcemingree<jJ5cat!on,quad-
rature, &c., which, on account of want of apace, cannot beinctudedin thia treatise. If Qhe the footof the perpendicular&om 0 on the tangent plane at any point J~ ît is easy to
Me that the aphere decribed on the diameter OP touchesthé locus of Q; and consequentlythe normal at any point Qof the pedal passes through the middle pomt of thé cor-
ïeopondingradius vector OP. It immediatelyfollowshencethat the perpendicular OR on the tangent plane at Q liesin the plane POQ, and makea the angle ~0~=jP(?~, Mthat the right-angled triangle QOB ts similar to POQ; andif we caUthe angle QOjR,a, so that the SrstperpendicularOQis connectedwith the radMMvectorby the equationp=p cos<[,then the secondperpendieularOB will be cos*a, and so on.~
It is obTioaathat if we form thé polar reciptocah of acurve or surface <dand its pedal -B,we ahallhâve a surfaceawhich will be the pedal of &; hence if we take a sar&ceand Its snccessivepedab the reciprocaiswill be
Thusthe radiu vectorto the pedalMof length«Kfe, andmakeawiththeradiusvectorto the curvethé angleua. UNagthMdeM-tionof the methodof denvattonMr.RebertahMeoMid~ed&ae<ica<dderivedoutMttand mt&MSM.Thus&?«" thecurvedetivedfromtheellipsele CNM:n!'<ovat. An mate~MMMï&eemaybe daM fromthee!):pM:tL
8718UKFACE8OEMVBDFROMQUADMCS.<
BB9
& sénés j8", ~8" .jS" the derived in the latter case
being negative pedats.It is atso obvionsthat the nrst pedal Mthe tKtWMe(Bt~~
JF%N)eCurves,p. 889)of the polar reciprocatof the given sur-face (that Mto say, tho surface derivedfrom it by aubatituting!n its equation, for the radius vector, ita reaproeat) and t!tatthe inverse of the series ~r,, will be the series
jS",S'
441. As we shall not have opportunity to return to thé
general theory of inversion,we give in this place the followingstatement (taken from Hirst, Tortolini,Vol. H., p. 165)of the
principalpropertiesof inversesurfaces.
(1) Three pairs of corresponding points on two inverseaur&ceslie onthe aameephere, (andtwo pfuraof correspondingpoints on the same circle) which enta orthogonallythé unit
sphere whose centre !s the origin.
(2) By tho propertyof a quadrilatéralinscribedin a circlethe Une ab joining any two points on one cnrve makes tbesame angle with the radius vector (~ that thé line joiningthe correspondingpointsa'y makeswith the radius vector M*.In the limit then, if ab be thé tangent at any point a-, the
correspondingtangent on the inverse carve makes the same
angle with the radius vector.
(8) In like mannerfor surfaces,two correspondingtangentplanes are equally inctined to the radins vector, the two cor-
responding normah lying in the same plane with thé radiua
vector, and &)rm!ngwith it an Moscetestriangle whoae baseis the intereepted portionof the ladins vector.
(4) It followsimmediatelyfrom(2)that the anglewhiehtwocurvesmakewitheachother at any point is eqna!to that whichthe inversecarvesmakeat the correspondingpoint.
(5) In like manner it follows from (3) that the angle whichtwo surfacesmakewith each other at any point iaequat to thatwhichthe inversesurfacesmake at the correspondingpoint.
(6) Thé inverse of a line or plane is a cirole or spheropasaing throngh thé origin.
8UMACB! DERIVED FBÛM QUADRICS.&72
(7) Any circle may be consideredas the intersection of a
plane, and a sphère through the origin. Its inverse there-fore is another circle, whieh ia a Mb-contraty section of theconewhosevertex is the origin, and which stands on thé givencircle.
(8) The centre of the secondcirde lies on the line joiningthe origin to a the vertexof the coneciroamscnbingthe sphère
along thé given cimle. For a M evidently the centre ofa ephereB which enta ~4 orthogonally. The plane thereforewhichia the inverseof cnts the inverse of B orthogonaïty,that Mto say, in a great cMe, whose centre is the aame asthe centre of B'. But the centresof B and of B' lie in a rightlinethrough the origin.
(9) To a circle otMoht&tgany carve, evidently correqModBa cMe oacotating the inverse cnrve.
(10) For niverae mrfaces, the centres of curvature of two
correspondingnormal eectionelie ina nght line with thé origin.To the normal sectionet at any point m correspondsa carvece aituated on a sphère paMingthrough the origin; andthe osculatmg cirele d of e' îs the inverseof c thé oscalatingcircle of a. If now t~ be the normal section whicb touchestt*at the point M', then by Mennier'stheoren~ the centre efc*is the projection on its plane of the centre of a, thé oscn-
lating cMe of al. But thé normalm'e, evidently touches the
sphèrej4 at so that <~is the vertex ofthé cônec!rcnmscr!bedto alongo', and theorem(10)thereforejbUowsfromtheorem(8).
(H) To the two normal sections at m whose centrea ofcurvatnre occnpy extreme pomlionaon the normal at m, will
evidently correspond two sections enjoying the Bame pro-perty therefore to the two principalsections on one surface
correspondtwo principal sectionson the other, and to a lineof curvatureon one,a une of cortatnre on the other.
w* e*442. The Sjst pedal of the eHIpsoid +P+ T='~
thé invemeof the reoiprocalempsoid,bas for its équation
BOBFAOa; DEMVBD MOM QTADMM. 87S
This eat&ceisFtemel'e Sur&ce of Ehst:ctty." The inverseof a system of confocalacntting at right angles ieevidently a
syatem of ant&ces of et<tst!citycatting At right angles; thelines ofcm~ture~erefoTe<)fthe<mr~<)fel~mtya!'eA~mmMdMdMi~MM~MBv~h~sm~Meof~e
eameMtaMdenved~mooncycKcqtMMbiot.The origin is evidentlya doublepointon thia aMt&ce,and
thé imaginary circle in which any spheMenta the plane <tt
infinity is a doubleline on the surface.
44S. Mr. Cayley firat obtained thé équation of the &mt
negative pedal of a qMadno,that la to eay, of thé envelopeof planes drawn perpendicnlar to the eeNtral tadii at theirextremidea. It is evident that if we desenbea apherepassingthrough the centre of the given quadric, and touching it at
any pomt dye', then the point a~e on the derived eut&eewh!t~ oomeepondsto a/y' ta the extrernity of the diameterof this aphœe, which paaeesthïoagh the eentm of the quadne.We <hnBeamiyfind the expresmoma
Now the secondof thèse equations is the differential,with
respect to <, of the Srat equation; and the required Mn'&ce
Mtherefore represented by the discriminantof thMequation,
SURFACES DEMVED FROM QUADBtCg.374
wMohwecan eaailyform, the equationbeingonly of thé fourth
degree. If we write thiabiquadratic
it will be fbund that and Bdo not containa!, y, z, while
Ct D, E contain them, each in the seconddegree. Now the
discriminantis of the sixth degree in the coefficients,and is
of the form ~+B' conaequentlyit can contain x, y, <!
only in thé tenth degr~o. This thereforeis thé degree of thesurfacerequired.
Its section by one of the principal planes consists of the&mt negative pedal of the correspondingprincipal section ofthe ellipsoid,whichMa cnrve of the sixth order, together witha conic,counteatwice,which is a doublecurve on thé surface.The double pointa on the principal planes answer to pointson th6 ellipsoidfor which !~+y+e"='8< or ?* or 2e*,aa
easily appears from the expressionsgiven for x, y, e in the
beginningof the artide. There is a cuspidalconioat infinity,and besides,a finiteonapidalcurveof the sixteenthdegree.
Thé reader will find (.P5&m~M<~a~MM<!c<MM,1858,and
Tortolini,Vol. H., p. 168)a discussionby ttr. Cayley of thedînèrent formaasaumedby the surfaceand by the cnspidalandnodal corvesaccordingto the dînèrentrelativevaluesof a*,y, d.
444. Mr. W. Boberts bas sdved the problem discnssedin the last article in another way, by provingthat the problemto findthé negative pedal of a surface,is identical with thatof formingthe equation of the parallel surface. The former
problemis to find thé envelopeof the plane
axB'+~+M'e''+y''+e"
wherea: y', satisfy the equationof the surface. Thé second
problem,being that of finding the envelopeof a sphère whoae
centre is on the surfaceand radius =Jb,is to hd the envelopeof
Now m finding this envelope the anaecented lettera are treated
as constants, and it Mévident that both proMems are particular
87bSURFACESMMYBD FBOM QUADNCB.
cases of the problem to 6n<t,under the aame cont~timm,the
envelopeof
And it is evident that if we have the equation of the paraUelsurface,we hâve only to write in it for i! a~+y*+< andthen ~c, ~y, ~e tora:, y, a; whenwehave the eqma~onof thé
negativepedaL Thushavingobtained by Art. 489the equationof thé parallel to a qaadno, we can find by the substitutionshere explained, the equation of the first negative, the engin
being aaywhere, as easily as when the origin ia the centre.
Further, if we write for A,k+k', and then make the eametmbBttto~onfor weobtainthe firat negative, thé origin being
anywhere,of the parallel to the qa&tbtc, a problem wbich it
wouldprobably not be easy to aolve in any other way.Having found,as above, the equation of thé first négative
of a quadrio,we have only to form its inverse, when wehavethe equationof the secondpositivepedal (Art. 44,0).
Ex. 1. To Cadthé envdepeof planeadrawnpetpendieabtiyat theextremitieseftheMdiivectorestothéplane<M!+ !y+M+d.
Herethe parallelsurfaceeoathtaof a pair of ptanea,whoseequationia(M+bytx +<~=?*,thatofiheenvelopeis therefore
( 876)a
CHAPTER XIV.
8T!MACEB0F THETHŒDDSGBBB.
445. THE general theoryof sar&cea,explained p. 190,&c.,g!ves thé followingreaulta, when applied to cubical sm'&ces.The tangent cone whose vertex is any point, and which en-
velopessacha surfaceia, in general, of the eîxth degree,havingsix cuspidal edges and no orduuuy double edge. It is con-
aequenûy of the twelfth clsaa, having twenty-&)aretattonMy,and twenty-seven doubletangent planes. Since then throughany une twelve tangent planes can be drawn to thé surface,any line meetsthe rectprocalin twelvepoints; andthetedpmcalM, in general, of the tweMh degree. Its equation can befound as at Bi~ef jP&Me<~OM,p. 99. Thé problem ia thesame as that of finding the conditionthat the plane
should touch the sor&oe. Maltiply the equation of the surface
byo',amdthenetlnunate&<ebythehelpoftheequa~onofthe plane. The result is a homogeneouscubic in a', y, e,containing alao a, ~3,'y, S in the th!rd degree. The d!sonmmantof thia equation M of the tweKth degree in ita coe&aents,and therefore of the thirty-aisth in ctj8y~:but this conNatsofthé equation of the reciprocal surface mnMpIied hy theirrelevant factor S". The form of the discriminantof a homo-
geneous cabical fnnction in <B,y, e ia 64<8"='T* (BS~ef P&t<M
C'MrpM,p. 190). The same then will be the form of the re-
dprocal of a mo'fMeof the thod degtee, S being of the fourth,and T of the aixth degree in a, /9, y, &; (that is to My,and T aje coM<MK)<M~Mm~of the given equation of thé above
degrees). It ia easy to see that they are also of the same
degree in the coemcîentaof the given equation.
BCRfÀCES0FTHETHtKDOMBEE. 877
446. Surfacesmay have either multiple pointaor multipleUnes. When a mr&ee bas a double Une of the degreep;then any planemeets the snr&oein a eecttonhaving p double
points. There M,therefore, the same limit to the degree efthe doublecurve on a Mu'&ceof the M**degree, that there Mto the number of double pointeon a cnrve of the degree.Smce a curve of the third degree c<unhave only one double
point; if a ear&ceof the third degree bas a double !ine,thatline muat be a right line.* A cubic having a double line is
neceamnty a rded anrface, for every plane passing thronghthis tine meets the Mr&ce in the double line, reckonedtwice,and m another line; but these other lines form a system of
generatomrestingon the doubleline as director. If we makethe doubleline the axis of < thé equation of the enf&cewillbe of the form
which we may write t<~+~+~=0. At any point on thedoublelino there w!l be a pair of tangent planes e'M,+c,=0.But as e' varies this denotes a syatem of planes in involution
( Cb!M<w,p. 887). Hence tXe~MM'of <<M~eM<~<!Me<<t<oKyjpoMt<on t~edoubleline, OMtMOco~<~a<eplanes o~a <~N<aMin in-00?M<Mtt.
There are two values of < real or imaginary, whichwillmake<«,+~ a pet&ct sqaMe; there are thereforetwo pointeon the double line at which the tangent ptanes coincide;and
any planethrough eitherofwhichmeete the surface!n a eecdon
having this point for a cusp. If the values of these squaresbe .X' and y, it is evident that M,and0, can each be expreMedin the form ~X*-tMy. If then we tam round thé axes so
If a eat&eehâvea doubleor other multipleline, the reciprocalformedbythemethod<~thelut articlevoaJdvaniehidentMaUybeeMMthemoxty planemeet9the am&tcein a earvehavinga doublepoint,andthereforetheplanear +py+els 8tDis to be consideredas touchingandthete<brethé phne '~+~4-'];<<<chtobe eoMidetedMtonchingthé mt&ee,mdependenttyef anytetationbetween ~9, <. 'HMM-ciprocelcanbeformed!ntbiscasebyeliminating«',y,t, wbetween« 0,<t"t<t,«~ 'V't <«,.
SCSfACES0F THETHHtDBMBEE.378
M to have for co-ordinateplanes, the planea JE, y, that !s to
Mty,the tangent planesat the cuapidalpoints; then every term
in thé equation will be divisibleby either a~ or y*, and the
equation may be reduced to the form )M!*=<e~In this form it is evident that the surface is generated by
lines y=\i)! <!M\'<c; interseeting the two directing lines a~,<Mp;and the generatomjoin the points of a system on <MC
to the points of a system in involution on icy, homographiewith the firstsystem. Any planethrough <!<omeets the surfacem a pair of right lines, tmd is to be regarded as touching the
aar&ce in the two points where thèse tines meet été. Thus
then aa the Une .cy!s a line, every point of which ma double
point, so the line été is a line, every plane through whieh isa double tangent. The reciprocalof this surface, which ie
that consIdeMdArt. 419, la of !ike nature with ItseKThe tangent cone whose vertex ta any point, and which
envelopes the Bmface,constatsof the plane joinmg the pointto the double line, reckonedtwice, and a proper tangent cône
of thé fourth order. When the point is on the double line thecono rednces to the secondorder.
447. Thera is one case, to whichmy attention waa oaUed
by Mr. Cayley, in which the reduction to the form <!a!'=M;y'is not possible. If u, and in thé last article, have a common
&ctor, then choosingthe plane represented by this for one ofthe co-ordinate planes, we can easily throw the equation of
the surface into the form ~-t-a:(M!+<oy)*=0.
It MhMeeupposedthat theplaneaJC,Y, the doubleplanesof theByetemin involution,are real Wecanalwaya,however,redaceto theform «'(.i~*)t2)H: the upperaigncon'espondïngto real, and thelower to ima~nary,doubleplanes. In thelatter casethe doublelinois altogether"KaUy"m the aar&ee,everyplanemeetingthe surfaceMa motionhavinj;the pointwhereit meetathe Imefor a realcode. Inthé formereaMthieis onlytmefora timitedportionof the doubleline,eectiomwhichmeetit elaewherehavingthepointofmeetingfor a con-jngatepoint thétwocuspidalpointamarkingtheseUmitsonthe doubleline. A right iine,everypointof whichlaa «Mp,cannotexiston acubiounteMwhenthesurfaceisa cone.
87&8UBPACB80FTHETHJRDDE&BEE.
The plane a; touches the eor&ce along the whole length ofthe doublefine,and meets the sarëtca in three coimoMentnghtUnes, The other tangent plane at any point coincideswiththe tangent plane to the hyperboloid«c+tfy. This case maybe conmderedM a limiting caae of that consideredin the last
article; vm, when the double director aiy coincideswith the
single one «x. The followinggeneration of the surface maybe given. Take a series of po!nte on a~, and a homographieseries of planesthrough it; then the generator of the cnMc
through any point on the line, Mesin the correspondingplane,and may be completely determinedby taking as director anyplane caMchaving a doublepoint where its plane meets thedouble iino.*
448. The argument which proves that a proper cabMcurvecannot have more than one double point does not apply tosnr&ces. In fact the line joining two double points, since itia to be regardedas meeting the surface in fourpomts, mustlie altogetherin the surface; but this does not imply that thésurfacebreaksup into othemof lower dimeamoM. The con-siderationof the tangent cônehowever suppliesa timit to thenumber of double points on any surface. We have seen
(Art. 251) that the tangent cone necessarily bas a certainnumber of doubleand cuspidaledges, and since every double
point on thé surface adds a double edge to the tangent cône,.there cannot be more double pointa than will make up thetotal number of double edges of thé tangent cone to themaximum nnmberwhich such a cone can hâve. Thus a curveof the sixth degreehaving six cnspa can have only fourotherdouble points; therefore since the tangent cone to a cubic isof the sixth order, having six cuspidal edges, the surface canat most have four double pointa.
When a surface bas a double point, the line joining this
point to any assmmedpoint is, as bas been aaid,a doubleedgeof the tangent cone from the latter point; and it is easy to
Thereaderiareferredtoammtetestin~geometn<!<JmemoironoubicalruledMr&cMbyCMmoaa,"Atte detRealeIstitutoLomb~rdo,"Vot.n.,p. 291.
SUBMCE90F TBBTBtBDDËOB&B.380
aee that the tangent planesalongthis doubleedge am the planesdrawnthiough thia!ine to tonchthe cône generatedby the tan*
gents at the doublepoint. If then this (Mmebreak up into two
planes, it followsthat sucha pointeataNsa cuspidaledge onthé
tangent cone tbmugh any assumedpoint A cubic then canhave only three such'biplanar double pointa. The rec!p)'ocaïof a cohio then having one or more double points may be
of any degree Ë'om thé tenth to the third, each ordinarydouble point redncing the degree by two, and each biplanar
by three.If the two planes of contact at a biplanar point coïncide,
the line joining this to any Msmnedpoint will be a t~tp~eedgeon the tangent cône through that point, and the degreeof the
reciprocatwill be reducedby six.
Ex. 1. Wtmth thedegreeof théreciprocalof.tyz =«~?P~t<M.Thereare threebiplanarpointain the plane<p,and the fee!-
procalis a onbio.
Ex.2. Wh&tlathereciprocalof'+~4.S~$f Pi)! tf t M
~<M.Thiateptetentaa caMehavingthe wtttcesof the pynmMd<~)Kefordoublepointaandthétee~mealmattbe of the&tatthdegree.
Theequationef the tangentplaneat anypointity~Wcanbefhfewn
intothefMm
+~+ 0, ~"M it &Uowathat the condition
that <t.t+~t<)<t<M eho<tMbea taegentplaneis
an equation whieh, cleared of nutiotb, fa of the fourth degree. GenetaHythe reciprocal of <H"ty + < tp< is of the form
(.B~<f .KetMOtH~, p. 102).A eabSchavingfour doublepointsh ahothe envelopeof
where a, e, e, l, M,« represent pJMee; 1 and«: /9:'y are two variable
parameteK. It !a obvions that the envelope is of the third degree; and
it la of the fourth ctttM since M we Mbetitttte the eo-MdimttM of two
po!n<<twe ean détermine four ptanet <~thé eyatem pasaing thMagh the
line joining thèse pointa.The tangent MM to this Mt&ee, whoM vertex le any point on the
surface,being of the fourth degree,and having fou double edget,mtMtbreak up into two oonesof the seconddenrée.
SORFACEa 0~ TBB TMRD DEQBEE. 881
44&.The equation of a caMchaving no multiple point maybe throwa into the form <M~+~'+e<'+<?a''+e«~0, where
a!,y, c, w represent planes, and wbere for )mnp!ict<ywe
supposethat the constants implicitly involved in a', y,&c.h&vebeeneo chosen,that the identical relation connectmgthe eqaa-tiOMof any five planes (Art. 87) may be written in the form
a:+y+<!+w+to=0. In fact thé general equation of the third
degree containstwenty tenus and therefore nineteen indepen-dent constants, but the form just written contains five termaand thereforefourexpressed independentconstants,whilebesidesthe eqaa~an of each of thé five planes implicitlyinvolvesthreecomtants. Thé form just written therefbM contains the sameaamberof constants ae the general equation. Thia &nn givenby Mr.Sylvester in 1851 (CbH~W%'eand .OttMtM~&<~Ma<M<
t&M~M~VoLVt., p. 199) is moat convenient for thé investi-
gation of the properties of cnMcalsar&oes in generaL*
460. If we write the equationof the Crst polar of any pointwith regard<oa sot&ce of the M**order
ïmM obaerved (JXt~ <~t~t Ath 18) that two fornu )My
~ppmentty eonttin the Mme number of independent ooMt~mta,and yetthat onemay be !eM general than the other. ThtM when a form Mfound
to contain the same number of constants M thé general eqmtion, it h
not <ttMh!tdy detnoMtt&ted that the general equation i< MdudMe to this
&??) and Ochteh has noticed a temarhble eMeption in the case ef cuves
of the fourth Mdor. ïh thé present CMe, though Mr. Sytvester gave M<tteotem w!th«t)t further demûMtMtion, he stttae that he WM in pomet-<!onof a pKof that the general equation could be rednced to thé sum of
&Tecubes and in but a ein~te way. 8ueh a proof haa been puMishedby Mr. Ctebteh (Ctw&, VoL ux., p. tM). He en-oneonaty aecribea the
theotan in the t~t to Steiner, who gave tt in the year t8a6 (CM~,VoL LDL. p. ÏM). It ohanced that Mt&eet of the third order werettedXd in this coantty a &w yeaM before German mttthemttticitmstamed
their attention to this tab}Mt< and eonMqaentty, thongh, M might be
expeeted&ont hh ability, M. Steineî't inveatigations led him to Mvertd
important MMttt, theM had be<n almoat aU weN tnown hete Mme yeambefoM.
SURFACES0F THETMtBODENREE.382
thea, if it have a double point, that point wtl satisfy tho
eqaattoaf)
where a, b, &c. denote second diffarential coeBScients eonw-
sponding to these letters, as we bave used them in the générât
équation of the second degree. Now if between the above
equations we eliminate ai'y'x'M',we obtain the loens of ail pointawhich are double points on tirst polars. This M of the degree4 (? 2) and is in fact the Hessian (Art. 864). If we eliminate
the ay-îte which cccor in <t, b, &c., since the four equationsare each of the degree (M–2), thé resuMag equation in icy~to'will he of the degree 4 (n 2)*,and will represent the locus of
points whose first polars have double pomte. Or, again, F Mthe locus of pointa whose polar quadrics are cônes, while thé
second surface, which we shall all < !s the locus of the vertices
of snch cones. In the case of snr<aces of the third degree, it
t6 easy to see that the four equations above written are sym-metrical between a;y~<eand a''y'<'<c'; and therefore that the
surfaces F and J are tdent!caL Thus then if <%<polar ~«K&'t<:
c~ any point with respect to <t cubic &ea cotM<cJ5<Mevertex
w B, the ~o&M*g!MaMcof B Ma coae tc~oeeweffKeM The
points A and J3 are said to he corresponding points on thepointx and B aro xaid to be correxponding pointx on the
Hessian (see ~i~ Plane C«n)M, p. 154, &o.).
451. The tangent p&MMto du B~MMmof ot cubio at i8<~ polar plane of B M~ M~eof ? <~ CM&M.For if we take
any point .4' consécutive to JL and on the Hessian, the poleof any plane tbrongh will be somewhere on the inter-
section of the nrst polars of and ji'; but these being con-
secutive and both cônes, it appears (as at B%'&efJP&MMCttn~
p. 165) that B, the vertex of this cone, is a pole of any plane
through and therefore of the tangent phme at A Andthe polar plane of any point on thé Hessian of a surface of any
degree is the tangent plane of the corresponding point B on the
surface J. In particular the <tM~M<t<planes to U along do para-bolia curve, <M*etangent ~&MM? the eM<tce J.' that is to say,
SURFACES0F THE THtRODEOREE. 3M
inthecaMofacabictAe <&oe!opa~~cM~MtiMcnM!~acMMca&M~Me~MMM<cCMnM)<t&o<e«m<CM&Mthe BeMMM.If
any line meet thé Hcssian in two correspondingpoints B,and in two other points C, D, the tangent planes atiaterMct along the line joining thé two points correapondingtoC,D.
462. We shall also invcstigate the preceding theoremabymeansof thé canonicalform. The polar quadrie of any pointwith regard to <+t~*+c.+d'e'+e)p' is got by eub~titutingforMits value (.K+y + e + v), whenwe can proccedaccordingto the ordinary ralea, the equation being then expreœed interma of four variables. We thus find for the polar quadrie<B'a!'+&+<j!+~'c'+eM'<c'~0. If we diRerentiatethis
equation with respect to x, remembering that <~<e==-<& we
get <?:)!='eto'M;and since the vertex of the cone must satisfythe four dinerendals with respect to fc, y, c, we find thatthe co-ordinatesa! y, i! e', M' of any point on thé Hessianare connectedwith the co-ordinatesx, y, e, c, w of B, thevertex of the correspondingcone, by the relations
<M:'a;=&~ =os'e= Jo'c ==ete'to.
And aince we are only concernedwith mutnal ratios of eo-
ordinates,we may take 1 for the commonvaine of thèse quan-
titiesana wite the w-orakates of BI1 1 1 1 1
titiesand write thé co-ordinatesof -B, t ) ~i< < –i
Sinee the co-ordinatesof B must satisfy the identicalrelation
a!+y+~+r+M'=0,we thns get thé equationof the Hessian
or &<!<&y<!<)M+ co~a~tea!-+ t&o~otM~+M~ctMy~+ a&e<n) = 0.
This formof the equation ehowsthat the line c<pliesaltogetherin thé Hessian,and that the point a~< is a double point on the
Hessian; and smce the five planes a*,y, z, e, <cgive nM toten combinationa,whether taken by twoaor by threes wehaveMr. Sylvester'atheorem that the~ee~aNM~~wt a~~oM~M<0~<Meten verticesare doubleJWM~on tlie B~MMtHand tC~Meten edgealie on <~ BeMtaM. Thé polar qnadne of the point
SURFACESOP THB THÏftD DESKEB.8M
aye Mdo'c'+ete'M', which resotvesitaelf into two planes inter-
secting along CM,any point on wMchEne may be regardedas the point B correapondingto osy~; thug then there afe <M
points toAoMpolar quadrics&MO&Mpintopairs 0/&MM<theae
pMKtsare <&«Mepoints on the JS~Mton,and the tK<~MC<t<M!<ejfthé <xwrMpoK<~tM~pairs o/*planes are ~MMon Me Hessian. It t
is by proving thèse theorema !ndepenJentty* that the reso-
lution of the given.equation into the sum of five cnbe6 c<m
be completely established.The equation of the tangent plane at any point of thé
Hessian may be written
453, If we coneiderall the points of a Sxed plane, their
polar planea envelope a snr&ce, which (as at BigAer JMmM
<X«~M,p. 152)is abo thé locnaof points whoaepolar qaadncaton<Athe given plane. Thé parametemin the equation of the
variable plane enter in the second degree; the problem îs
therefore that considered(Ex. 3, Art. 448) and the envelopeia a cob!c mf&eehaving fonr doublepointe. Thé polar planesof the points of thé sectionby the cubioare the tangent ptaneaat thosepointe, consequentlythia polar cubic of the given planeis inscribed in the devolopableformed by the tangent planesto the cubic along the section by thé given plane (J9%rAe)'
It ~nHa~est &omtheappendix"on théorderof systemof eqoa-6MM,"that a tymmettiedetennimmtof p M~~and eotumn~eaoheon-stituentof whiohlaafimct!onof thett"'ordertnthe ~DaHes,tepteMntaa <n)f&Meof thé opdegreehavingtf(?* 1) doublepoints}and thusthattheHeMianoft Mt&oeoftheM"degreealwaysbastO(a 2)*double
poiate.
SURFACESQFTHE THtRDDEGREE. 385
ce
Plane CMtMa,Art. 161). The polar plane of any point A of
the section of tho Hesaian by the given piano, touches the
HesHan (Art. 451) and is therefore a common tangent planeof the Hessian and of the polar cubic now under con-
sidération. But thé polar quadric of 2?, being a cone whose
vertex is A, is to be regarded as touching the given planeat hence B is aiso thé point of contact of this polar
plane with thé polar eubie. We thas obtain a theorem
of Steîner's that polar cubic of any plane <oxc~ <~
Hessian along a certain CMrce. This curve is the locus of
the points B corresponding to the points of tho section of
the Hessian by the given plane. Now if points lie in any
plane ~4-<My-t<M+~)c+gw, thé corresponding points lie on
the surface of the fourth order–+~-+1+ ~.+-2-.
Now<M!! cy M <<t) fM
the intersection of thia surface with the Hessian is of the
sixteenth order, and includes tho ten right lines xy, ZM,&c.
The remaining curve of the sixib order is thé cnrve alongwhich the polar câble of the given plane touches thé Hemian.
Thé four double points lie on thia curve; they are tho
points whose polar quadrics are coaes touching the given
plane.
464. If on the lino joining any two points a:y.e', ic'y~we take any point a!'+Xa: &e., it is easy to see that ite
polar plane is of the form ~,+X~ where J~are the polar planes of the two given points, and J~, is the
polar plane of either point with regard to the polar quadricof the other. The envelope of this plane, considering X
variable, is evidently a quadric cone whose vertex is the inter-
section of the three planes. This cone is clearly a tangentcone to the polar cabic of any plane through the given line,the vertex of the cone being a point on that cubie. If the
two assumed points be corresponding points on the Hessian, JP,vanishes identically; for, the equation of the polar plane, with
respect to a cône, of its vertex vanishes identically. Hence <~
polar plane of any point of <~ ?ttMJoM~Mi~<<eocon~poM~M~
~K<< on the J2eMMM~<MM<~<'0t<~&the <K~~eC<<ÛMof the ~M~<
SURFACES0F THE THtBD DMMEt386
planes to the &?&? at <<«?p<wt~ In any assamed planewe ou drav three Unesjoining corresponding pointa on the
Hessum; for the carve of the aixth degree cmxMeredin the
laat article meeta the assmnedplane in three pairs of con'e-
epondingpointe. The polar enHo then of the aasnmedplanewill contain three right lines; ag will othorwiseappear &oMthe theory of right tmeson caMcswhieh we shallmowexplain.
4M. We said, note, p. 39,that a caMcal surfacenecMMmIycontauMright linea, <mdwe now enquire how many in générâtlie on the enrfMe.t In the first place it Mto be observedthatif a right line tte on the surface, every plane throngh it ia adouble tangent plane becanse it meets the surface in a rightMneand conM; that is to eay, in a section having two doaMe
points. The planes then joining any point to the right lineson the surface are double tangent planes to the sar&ce and
thereforeaiso doubletangent planes to the tangent conewhosevertex ie that point. But we have seen (Art. 445) that thénumber of aach double tangent planes M <toet!~set!eK.
This resnlt may be otherwiae establiehed as follows: letun suppose that a cabic contains one right line, and let us
examine in how many ways a plane can be drawn throaghtbat right !!ne, auch that the conio in whieh it meets thesor&ce may break up into two right tines. Let thé nghtline be <<?; let the equation of thé annace be tp!y'=oP'; letus substitnte <o=~«!,divide out by <, and then form the dia-cnminant of the KsaMng qoadric in <~y, z. Now in thia
qaadrto it ie aeen without dMScoltythat the ooeScients of
a! ay, and only contain in the firsi degree; that those of
Steinersayt that thereeMonehnadted linee'aeh that the polarplane<f anypointef onecf thempaMeathrongha &tedUne,but Ïbelievethathb theoremoughtto be<nnendedMabew.
t ThetheMyef rightBneton a enMcatMr&ee~aa SKtatudiedinthe year1M9ima ecn'eep<mdeMebetweenMr.Cayleyendme,théMMl<aof whichwerepahMahed,Ctmtt~ <M<J)«Mt ~MtoM~~ AM~,VoLtv., pp.lt8, M& Mr.CayleyaMtctMarMdthat a fte&tttenamhwof rightlinostM<tHeontheaM&te;thé deteminaRenof that nambefa*above,andthé (ËaenMteminArt 4Mwereeuppliedbyme.
BUM'ACBSOP THE TBIBP CEtUtEË. 887
CC2
CM?and y< contain in thé second degree, and that of e* inthe third degree. It followshence that the equation obtained
by equating the discriminant to nothing is of the nfth degreein and therefore that ~MM~ any Wy~<&tMon a cMMxt?
<M)~MeMMbe <~MKfive planes, each 0~tcXM&M<e~Me«ti~Me<K<!KO<~pair of rig"t M! and coaseqaently every~/t<line Mta <MM!& M<N'MC<e<?by <eK0<&<M.CoMtder now thesection ot the surface by one of the planes juet referred to.
Every Uneon the sm&cemuet meet in Mme point the section
by this plane, and therefore must intersect Mme one of théthree lines in thia ptane. But each of these lines is inter-aected by eight in addition to the linea in the plane; thereare therefore twenty-four lines on the cabic besides the threein the plane; that is to say, <<MK<!y-MMttin <t~.
We shall hereafter show how to form thé equation of asurface of the ninth order meeting the given cubic in those
Unes.
466. Since the equation of a plane contains three inde-
pendent constants, a plane may be made to Mnl any three
conditions, and therefore a nnite nnmber of planes can bedeterminedwhich shall touch a surface in three pointa. Wecan nowdéterminethis number in the case of a caMcalsor&ce.
We have seen that through each of the twenty-seven linescan be drawn five triple tangent planes: for every planeintefsecting in three right Unes touches at the vertices of thé
triangle formed by them, thèse being double points in the
section. The number &x 27 is to be divided by three, s!nce
each of the planes containsthree right lines; <~<Mare <~M~Min «Hj~y~oe <~& <aHyeK<~&!tMN.
4B7. EveryF&Mte<w~& a right Mue<?a c<<M3Mo6wtM~a <&W&&<(tKyeK<plane; and ~Mt!r<of pointa < <!<M!<tM<~fWt? <y<<6Ntin involution. Let thé axis of z lie on the aorface,and let the part of the équation which is of the firat degreein:c and y be (<M!'+~+<')a;+(o'<+yj:+c')y; then the two
points of contact of the plane y=~a' are determined by thé
equation
SURFACES 0F THE THÏBD ÏHBMtEN.888
but thit denotes a Systemin involution (C'<M«a,p. 28?). Itfollowshence, from thé known propertiea of involution,thattwo planes can be drawn throngh the Une to touch the surfacein two coincidentpoints: that Mto say, which eut it in a lineand a conie touching that Une. Thé points of contact are
evidently thé pointswhere thé right line meets the paraholioeurve on the surface, It was proved (Art. M6) that the rightline touches that curve. The two pointa then where the Mnetouches the parabolic cnrve, together with the points of
contact of any plane through it, form a harmonie system.Of course the two points where the line touchesthé parabouccurve may be imaginary.
4ë8. The number of right lines may atso be determinedthus. The fonn <Me=~ (wherea, &e. repreaentplanes)is one whichimplicitlyinvolvesnineteen independentconstants,and therefore is one into which the general equation of a
cubic may be thrown.* Th!s surface obvioady containsnineUnes (ab, e<f,&c.). Any plane then 0'='~ which meeta thesurface in right Unesmeeta it in the same lines in which itmeets the hyperboloid~tee='<~ The two Hnesare therefore
generatoraof differentspeciesof that hyperboloid. Onemeets
the Mnesed, ef; and thé other the Unes cf, <&. And, since
/t bas three values,there are three unes whichmeet ab, cd~if.The samething followsfrom the considérationthat thé hyper-boloid determined by thèse lines must meet the surface in
three more Unes(Art. 818).Nowthere are clearly six hyperboloids,<tt,<~ef; < c~ de,
<&e.,which determine eighteen linea in addition to the nine
with whieh we started, that is to say ae before, twenty-sevenin ait.
If we denote eachof the eighteenUnes bythé three which
it meets, the twenty-eevenUnesmay be enumeratedas follows:
there are the ongnMtInine ot, ad, o~ c&,cd,<~eb,e<~< to-
gether with (<c~.e/'),, (a~.c~ (a~.c~),, and in litemanner three Unesof each of thé fbrms <<&, <td'.tc.e~
It wiUbe found in one hundred and twentyWttyt.
SURFACES0F THETHUU)MGBBE. ?9
ad.be.cf, a/&c.<&,<&e.< The 6ve planes which can bedrawn through any of thé Unesab are the planes a and b,meeting reapectivelyin the pture of lines ad, af; ~o,& andthé three planes which meet in (<!&.eaL~'),, (o&.c~.<&),;(a&.<e~),, (<t&.c~.<~),;(aA.c~ (a~c~e),. Thé five
ptaneswhichcan be drawnthroughany of the lines (a6.<'d'),,eut in the pairs of lines, ab, (<c/.<& o~, (<t/c~),;
(o~c.~),; and in (<Mf.&e.~),,(<t~Ac.<&),;(a~),,(«/d!e),.
459. Pro& SehSR! haa onde & new arrangement of theHnes(QuarterlvJournal o~ ~a~~ma~eft,VoL 
890 SURFACES0F THE THtBD DEQKEE.
460. We can now geometncatîy comtract a system of
twentyseven lineswhich c<mbetongto a caMcaleai&ce. Wo
aM~y8tta'tbyta!dngtu:bit!'ar!lyaay~ne<t,aad&veo~et9which interaect it, & These determine a cnbieal
surface, for if we descrtbesach a Bnr&tcethrough four of the
points where a, is met by thé other Unes and through threemore pointson each of theM lines,then the cubic determined
by these nineteen points oontainaa!l thé Une~~nce each linebas four points conuaonwith the surface. Now if we are
given four mon-mtersec~nglines, we caa in general draw twotraBsveta&bwhichA&!1intersect them all; for the hyperbo!otddeterminedby any three meete the fourth in two points thronghwhich the trMMverMiapaaa.* Throagh any four then of theKnea we can draw in addition ta the line o, anothertransversalo,, whichmust a!solie on the saf&ce since it meetait in four points. In this manner we conetrnct the five newlinea «~ o~ an o~ < If we then take another ttansvetsd
meeting the four first of these lines, the theory tth'oadyex-
plained shows that it will be a Ime which will a!someet
If thetypetM~dtoucheathe&mfhHae,thétwotauMwmohfedoeeto a singleOM,andit it évidentthat thehyperboloiddeterminedby anythreeotheMofthe&MrlinesabotouoheatheMmMningone. Thiaremark1 bélier is Mr. Cty!ey's.If wedenotethe conditionthat twolineashouldmtetMetby(M), thenthé conditionthat four lineaehoaMbemet by onlyone tMmvetMdia e~pfeMedby equatingto aothia~thedeterminant
The wnMdng of the detenninantformedin the Mmemanner fromnw
linea, h the conditionthat they are all met by commcn tMMtetMLThe vMitMBg'of the timilMdeteradnant for dx lines, MpteMta that
they are eennectedbya reMon whichhMbeenMNedthe iavoluttonofaix !ine*t" and whichwillbe MtMedwhenthe lines<tMbe thédheeti~Mof mt foreMin eqaiNhtiam.The reader wiU &td eeveMi intetesthtgCMMMmct&MMon <M<aubjectby MeMN.Sylvesterand 0&yky,and byM. ChMtM,in the ChatptMJBM<6Mfor t89!, Prmier &M«<f<.
891SUM'ACES OP THE THKD DMBEB.
tho SMt. Wo have thns constracted a "donMe-six."Il Weeean then immediatelyeonetrcct the remaining lines by takingthe plane of any pair «~ which will be met by the lines
a, in points whieh Heon the line o,<
481. M.MtaSt has made an analysisof AediSerent
species of oabics aoeordingto the MaHtyof thé twenty-sevenlines. He &<dathus nve spec!es: A a the lines and planes
real; B, fifteen lines and Meen planes real; C. <evenlines
and five planes real; that is to say, there is one right Kne
through which five real planes caa be drawn, only three of
whichcontainreal triangles; D. three linee and thirteanplanesreal namety,there is one real triangle through every aideof
whieh pass four other real planes; and, B. three lines and
eevan planes real
1 have aho given (C~tKM~a and D<<t~ JMtt<~<Ma<&<~
J<M(HM~Vol. IV., p. 266) an enumeration of the modifications
of the theory when thé surface haa one or more doaMepoints.It may be attted genemlly that the cnbîo has aiwayatwenty-seven nght lines and &)rty-nvetriple tangent planes, if we
count a line or plane through a double point as two, thronghtwo double pointa as four, aad a plane through three anch
points as eight. Thus, if the surface bas one double point,there are six lines pattsing throngh that point, and fifteen
other lines OMin the plane of each pair. There are fifteen
treble tangent planes not passing through the double point.Thns 2x6+15=87; 8xlo+16=.46.
Again, if the surfacehave four double points, the lines are
the six edgesof the pyramid formedby the four points(6x 4),
together with three o~em lying in the same plane, each of
whichmeets two oppositeedges of the pyramid. The planesare the plane of these three lines 1, six planes each throughone of theae Unes and through an edge (6x 2), together with
the four &ce!)of the pyramid (4 x 8).The reader will findthe other cases diMumodin thé paper
just te&ned to.
INVARIANTSAND COVARtANTS0F A CUBIC.3&2
tNVABïANTSANDCOYARIAKTS0F A CCBtC.
462. Wo shall in this section give an aceotmt of the
principal invariants, covariants, &c. that a cubic can hâve.We only suppose the reader to have learned from the Z<Mû<Mon Higher Algebra, or elsewhere, some of the most elementary
properties of these fonctions. An invariant of the equationof a surface is a function of the coefficients, whose vanishing
expresses some permanent property of the surface, as for
example that it has a nodal point. A covariant, as for
example the Hessian, dénotes a Bur&oehaving to thé originalsurface some relation which is independent of the choice of
axes. A contravariant is a relation between a, ~8, 'y, S,.ex-
pressing the condition that thé plane aai+~+'ye+Ste shaH
have some permanent relation to the given surface, as for
example that it shall touch the surface. The property of
which we shall make the most use m this sectMn is that
proved (ZeMOMon Higher .~e&f< p. 66), viz., that if we sub-
atitute in a contravariant for a, &c. &c., and then
operate on either the original fonction or one of its covariants
we shall get a new covariant, which wiU reduce to an invariant
if the variables have disappeared from thé result. In like
manner if wesnbstitute in any covariant for a', y, &c. ?- )j~t
&c., and operate on a contravariant, we get a new contravariant.
Now in discussing thé properties of a cabic we mean to use
Mr. Sytvester's canonical form in which it is expreased by the
sum of five cubes. We hâve calcnlated for this form the
Hessian (Art. 458), and there would be no dimcalty in calca-
lating other covariants for the same form. It remains to show
how to calculate contravariants in the same case. Let as
suppose that when a jfunctionU is expressed in terms of four
independent variables, we have got any contravariant in a, ~8,
<y,8 and let us examine what this becomes when the function
is expressed by five variables connected by a linear relation.
But obvioosly we can rednce the function of five variables to
one of four, by suhstitating for the fifth its value in terms
JNVABtANra AND COVABtANTS0F A CCBtC. 898
of tho othera; vtz. M<=-(ai+y+e-t-t)). To find then thecondttMnthat the plane <KB+ + + 8e+ e<emay have anyMsignedrelation to the given eor&ce,Mthe same problemasto find that the plane (a-e)a!+(~-e)y+(Y-e)e+(~-e)o v
may have the same relation to the sur&ce,!t8 equation beingexpressedin termaof four variables; eo that the contravariantin five letters îa denved from that in four by aubstituting
a-e, ~S-e, 'y–e, ~-e respectively for et, '1, 8. Everycontravanant in five tettem is therefore a funetion of thedifferencesbetween a, y, e. This method will Bebetterundemtoodfrom the followingexample.
Ex. Théequationofa quatMolagivenin thefoim
where <+yt<n'+w0: toand the condition that M-t~y+~t'~+'w
tMy touch the sur&<!e. If we reduce the equation of the quadric ta aOmetienof four vanaMea by mbttitudng for tf ïta value in terme of the
otheK,the<!oeaeietttsof.t',s',z',e'aKrMpeetiTe!yaK, t~, eKt, ~teewhile evety other ooeSeient becomes<. It now we substimte fheM ~!ue<
in the equation of Att 76, the condition that the plane <t.t+~y+'~+ ?touches, becomes
46S. We hâve referred to the theorem that when a con-
travariant in four tetten M givet~ we may sabsdtate for
«,~9,y, S dM~ïenti&taymbolswith respect to <c,y, i! <o;and
that then by operating with the fonction so obtained on anycovariantweget a newcovariant.. Supposenowthat weoperateon a function expreasedin term9 of five letters a~y, <c.
Sinec <eappem in this function both explicMy and atse
where it M introdnced in <c,the dMerentîat with respect tod d dio J!th 1 t,iB M-T-
+ -?- or, in virtae of the relation connectmgMaa! aM a.<c
INVAMAMSANDCOTANAttTSOPA CUBÏC.?4
w!th the othervMMtMee, HenM&contïwvMiamtin
&nrlettem!etomedmto<mopera.tmg6ytnb<)l in Sveby
Ntb8t!ttltmgfor
But we have aeen in thé tMtM'tidoth&ttb6cont!'Kva.tMatin five tettembas been obtained from one in &nr, by writingfor a, a-6, &c. It Mkws theu unmed!&tdythat in any<!Mt<MttKtf«M<Mt~M ~'MM'<we <M~S<&«<6for &, S, a,d d d d d _1' '_1.1 .1
<S'<S'3~'<M~«~ <M < «~
«'AM&OpeMt~ <Mttlte <M~Mt<tJ'~MKe<<!)'(Won <tMy<!0<M~M)t<,we obtain <t <?<«<!ooa~«tH<M*Mt<wrMM<-The importance ofthtsis&atwh~weh&veonM~unAaMntrtvatMmtoftheform in five letters we eu obtain a new covariant withoutthe tàbor!onsprocessof reoarring to the form in four lettem.
Ex. Wehave<eenthatBe~(a j9)*Ma contMvatiantof theform
Ifthen weopeMteonthe qtMdnowith~~(~"s-)
,<heKsdt,~Meh
<m!y dM~m by a anmetie<d &etûr &Mn
~h+~M+~~+M~+~~
h aniavariantcf the qaaddc. It fa in fact itadieedminant,and eoaMhavebeenobtained&omthe expreMhmArt.63,by~titîag Mtn the htt<utMe<~e, t+<, e+4 <<+<a for a, d, and pttttin~<t!lthé other<!MSMentseqcalto<.
464. Lt like jmanner it is proved that we may aabsûtntein any covariantfonctionfor !C,y, <, wi (M~rentud Bymbobwith regard to ft, y, 8, e, and that operatingwith the functionso obtained on any M&t)'&van<mtwe get a new contravariant.In fact if we Sratreduce the fanction to one of four variables,~<;h<Bmm~ethedMïeTentMtMh~tat!on which we have a
r!ghttodo,wehave<<nbatIt)KteA&r
INVARIANTS AND COVABïAt!T8OF A CUBIC. SM
But ainee thé contravariantin five lettem was obtainedfromthat in four by writing et-e aforet,&a. it iB evident that thedMFerentmtsof both with Mgmd to a, y, S are the eame,while the diferential of that in five tettem with respect to a!a thé aeg&tïve aam of the dMarentM.taof that in four letterawith respect to a, /9, 'y, S. But this establisheathé theorem.
By th!a theorem and that in the laet artide we can, beinggiven any covariantand cmtttavMMmt,generate another,which
again combmect with the &nnef givea t!ee to new oneswithout Umit.
466. The polar quadrio of any point with regard to thecabio a!i~'+<M*+<~+~ M
Now the HeoeMmM the d!scnmimantof the polar quadric,Its equation therefore,by Ex., Art. 463, is S&<!(!eyj:txo=o,aawM aheady proved, Art. 452, Aga!m,what we have caHed
(Art. 468) the polar cubic of a plane
MagthecondMonthatthîaphnesho~ddtonchthepoiarquadnc is (by Ex., Art. 462) Se&M)tp(et-~)''=0. This Mwhat !s caUed a mixeAconcomitant, since it conttuna botheeta of variables fB, &c., and ?) /9, &o.
If a<w we BtdMtIttttein tHs for a, &c., &o.,
and operate on the on~aal cubio,we get the HessMn; but
ifweûperatoontheHesa~wegetacovariantoftheËf~order in the vanàbtM, and the seventh in the coe~aents to
wMohweaha~a&erwardaM&t'aa~,
In order to apply the method indicated (Arts. 463,464) itis neceMMyto have a contraTmiant; and for thia purpose1
havecàtcalatedthe contravariant<r wMchoccnrsin the equationof the MCtpMoalam'&ce,whicb, as we have already seen, iaof the form 64<~=T~. The contravariant <r expresses thé
ÏNVAMANT8ANDCOVANANTS0F A CUBtC.396
conditionthat any plane aa!+~+&c. shouldmeet the surfacein cubic for whieh AronhoM'sinvariant S vanishes. It iaof thé fourth degree both in a, ~3,&c. and in the coefficientsof the cubie. In the case of four variables the leading term
is d' multipliedby thé of the temMtycubic got by making<p'='0in the equation of thé surface. Thé remaining termsare ca1culatedfrom this by meam of the diffmntial equation
(ZeMo~ M Higher ~J~e&Mt)p. 70). Thé form being foundfor four variables, that for five is calculated from it as in
Art. 468. 1 snppres: the details of the calculation whieh
though tediousprésenta no dintculty. The resuit is
For taciHty of reierenee 1 mark the contravariants withnumbers betweenbracketaand thé covariantsby numbers be-tween parenthèses,the cubio itself and the Hessian beingnumbered (1) and (2). We can now, as already explained,from any given covariant and contravariant generate a new
one,by subatitutingin that in whichthe variablesare of towest
dimensions, differential symbolefor thé variables, and. then
opMatmg on the other. The result Mof the aifferenceoftheir degrees in the variables,and of the sum of their degreeain the coefficients. If both are of equal dimensions,it is in-dinerent with which we operate. The result in this case isan mvanamtof the sum of their degrees in the coenîotenta.The results of this processare given in the next article.
466. (a) Combining(1) and [l], we expect to find a con-travariant of the nrst degree in the variables, and tho nfthin the coemdents; but Unsvaniahes identically.
(~) (2) and [1] gives an invariant to which we shati re&rM invariant ~t,
=. B~c'~ SoMeSo~c.
If A be expressed by thé symbolicalmethod explained(ZeMMMon J5~~ ~~e&Mt,p. 77), its exprosmonia
(t2M) (1246)(1M7)(8848)(6678)'.
897mVAMANTa AND COVABtANM OP A CUBÏC.
(~) Combiningwith (8) the nuxed concomitant(Ait. 466)weget a covariantcubicof the ninth order in the coeScienta
<!&c<&3c<&(a+6)ecto.(6).
(h) Comhmmg (5) and [1] we have a Unearcontravariantof the thirteenth order, viz.
It seemaTumecesaaryto give further détails M to the steps
by whichparticular covartamtsare &~md,and wemay therefore
sum ttp the principalrésulta.
467. It is easy to see that every invariant is a symmetricfnnctton of the quantities <~b, c,J, e. If then we denote the
sum of these quantities, of their prodncts in pairs, &c., by
p, f, a, t; every invariant can be expressed in temMof
thesefive quantities,andthereforein terms of the fivefollowingfundamentalm~ariants, which are all obtained by proceedingwith thé processexemplifiedin the last article
We can, however, form skew invariants which cannot be
rationallyexpromedin terms of thé five fnndamenttttinvariants,
although their squarescan be rationally expressedin termeof
INVARIANTSANDCOVAMANTSOPA CttBtC.898
thèse qnactttiea. Thé e!mp!eatinvariant of this Hnd is got
by expresaing in terma of ita coe<Bctentsthe discriminant
of the equation whoaeroote Me a, c, <~e. This, it will
be found, givea in terma of the fundamental invariants
B, C,J9, an expresmonfor multipliedby the productof thé squaresof thé tMeMnoesof all the quantitiesa, b, &c.
Thia invariant being a pertect square, its square root is an
invariant F of the one hanJtedth degree. Ita expression in
tonus of the fundamental invariante is given, jPAt!Mt}p~M<~!R'<M!Mc<«MM,1860,p. 23S.
The discriminantcan eamiybe expreasedin tenM of thefundamental mvantmta. It is obtained by etimtnatmg the
variablesbetween the fourdiNeren&tbwith respectto a!,y, x,that !Bto aa.y,
H<mce~,y*eMpropordonalto6c<~cf!e<t,&e. Snb-
sdtuting then in the equation!c+y+z+o-Ko'=0, we get the
(ttBcnmmamt
deanng of radicale, the result, expressed in terme of the
principal invariants, i9
468. Thé cuMch<a&Ha'6mdamMtalcovar!<Hitplanes ofthe orders11,19,37,48 in the coeSMenta,~]z.
Every other covariant, indndmg the cubic itad~ can in
général be exptesaed in tenas of thèse four, the coe~o!ent8
being invariants. Thé conditionthat thèse four planes abouldmeet in a point, ia the invadant F of the one hamdredth
degree.There are linear contr&vanantathe s!mpl<6tof which,of the
thirteenth degree, haa been already given; thé next being ofthe twenty-Smt, <*S(<)(a-/8); the next of the twenty-
ninth, <'Sc&(<t-t)(et-/S),&c.
ïNTABtANTS AND COVARtANM 0F A CDBtC. 899
There are covariant quadricsofthé Nxth,&na'teen&,twenty-second,&c.ordera; and contravariantsof the tenth, eighteenth,&c.the ordermcreMtngby eight.
There are covariant cnMcaof the ninth orderS<c<&(<t+~)<Me,and of the sevonteenth,~So~a: &c.
If we caMthe ongtMt cubic U, and this bat covariantmnce if we form a covariant or invariant d Ï~+\P, thécoeiBdentsof the severalpowers of Xare evidentlycovariants
or invanants of the cubic: it followsthat given any covariantor tnvanant of the caMcwe are discussing,we can form fromit a new one of the degree sixteen higher in the co<tSdents,
by performingon !t thé operation
Of highercovariants weonly think it neceaeatyhere to mentionone of the SMi order, and SAeenth in the codBc!ents~Mwwhich gives the five 6mdMnentaïplanes: and one of thenmth order, 0 the tocneof points whose polar planes with
respect to thé Hessian touch their polar q<Mdncawith teepectto U. Its equation is expreasedby the determinant at the
top of p. 60,if a, ~9,&c.denote the 6rst diSerentialcoeSdentftwith respect to the HeaMsn,and a, 6, &o. the secondd!Se-rentials with respect to the cttMc.
The equation of a covariant whoae mtemectt<mwith the
given caMc determinea the twenty-seven KneBis 0=:4H~where hMthé meaning explained,Art. 466. We sha]!giveM. CkbBch'sproof of this at the end of the volume. 1 had
vennedthéform,which hadbeenauggestedto mebygeometricalconfdderatMmB,by examiningthe followingform, to which thé
equationof the cubic canbc reduced, by taking for the planée<cand y the tangent planes at the two points where any !memeets the paraboliocurve,and two determinateplanesthronghthesepointsfor the planes <p,<
The part of the Hessiamthen which does not containeither<eta'y !B~M~: thé comeepondiDgpart of <' M-2(a~+oh~),and of 0 ia -8<c~(<~+<~). The euï&oe 0-4H~ haa
INVARIANTSAND COVANANT8 0F A CUBTC.400
therefore no part which doeenot contain either a) or y, ana
the line a~ lies altogetber on the surface, as in Uke manner
do the rest of the twenty-seven Unes.
Thia Mctica h ahridged &cnt e paper which 1 contribated to the
PA<!M~)iMoa<2~<MM<t<*<MMM,1860, p. 229. Shoftty after the reading of
my memoit, and before it~ puMMia~cn,there appeared two papert in
CreUe'a J!Mfn«~ Vol. M, by ~-o&tto)- CtebMh of CN-htahe, in whiohtome of my resulta were Mtieipated: m particular the expression of all
the în~Mtiantaof a cubio in term of aw thndameatal attd the expreMongiven above for thé tut&ee paMing through the twenty-teven lines. The
method however whieh 1 pnKned wM different from that of ProfetMt
Ctehseh, and the dheMden of the covariants, as well u the notice of
the invariant F, 1 Mieve were new. Clebaoh haa etpMMed his laati!o<MiBWMnta aa &n<tioMof the coefficientsof the HeMitn. Thus thetecond is the invariant (12M)' of the Hetaian, &o.
( 401)
DD
CHAPTER XV.
GENERALTHEOKTO? SURFACES.
469. WE shall in thia chapter proceed, in continuationofArt. 256, with the general theory of sur&cea, and shall nratmention a &w nusceltaneoas theorema which are Mmei~meeusent!.
2%<&M<Mq~' Me~m~to&Me polar ~&N:~ with ~aM? <0
four MO~MM JV, Q (M~OM<J5~WMare M, M, <~?!<?<
<K<tpMK<,t9«~M)~!tce~ed!~)'eem+«+jp+g-4. Forits
equationis evidently got by equating to nothing the determi-nant whose constituents are the four dMerentM coefficientsof each of the four snr&ces. If a surface of the formoJtf+~+cJP touch the point of contact is evtdentty a
point on the locua just considered, and must lie somewhere
on the carve of the degree g(~+M-t-~+g–4) where is
metbythelocaasur&ce. InIikemanne)*)~(at+K+p+g–4)snf&ceeof the form <~+ bN, can be drawn eo as to touchthe carve of intersection of J~ Q; for the point of contactmust be somoone of the points where the curve PQ meeta
the locnssurface.
It &l!owa hence that the condition that two of the
mp points of intersection of three snr&cea JM, P maycoincide, contains the coenidenta of the nmt in the degreeKp(8M+M+~–4); and in like manner for tbe other twosar&ces. For if in this condition we sabstitnte for each
coeciciento of ot+Xa',where a' is the correspondingcoefli-
cient of another surfaceAf' of the same degree as M, it ia
évident that the degree of the result in is the same asthe number of surfaces of the form J~+\Jtf which can hedrawn to touch the curve of intersection of N, -R*
Moutard,2~~MM'<~MtM&<,VoLx!x.,p. M.
GEMBNALTHEORT 0F SURFACES.402
1 had arrived at thé same result otherwise thoa: (see
()M<M-<e~JOttt~M?,Voh p. 339) Two of the points of inter-
section coincide if the curve of intersection 3f<Vtoach thecurve MP. At the point of contact then the ,tangent planesto the three BNJtaceBhave a line in common: and thèse planestherefore have a point in common with any arbitrary plane'xc+~+<~+<?«'. The point of contact then eatisfies the
determinant, one row of ~Mch is a, c, d and the otherthree rowa are formedby thé four differentiaisof each of théthree snr&ces. Thé conditionthat this determinant may be
sa~Bnedby a point commonto the three surfaces is got byeliminating betweenthe determinant and P. Thé rescitwill contam a~b, c, d in the degree mnp; and the coefficientsof M in thé degree )tp(w+M+~–3)+MMp. But this resultof elimination contains as a factor the condition that the
plane aa)+i~+M+<?<e may pass through one of the pointsof intersection of JR And this latter conditioncontaina
a, 6, c, d in the degree mnp, and the coeBBdenteof M
in thé degree np. Dividing ont this factor, the quotient, as
already seen, containsthe coefficientsof M in the degree
470. The locus of points whose polar places with regardto three surfaceshave a right line common, is, as may bein&rred from the last article,the cnrve denoted by the Systemefdetennmamte
where «“ &c.denotethe dinerentialcoeScients. But thia carve
(seeAppendix)Mof theorder (M"+ M"+p"+ <M'M'+M'p' +~'w'),wheM M' !a the order of «“ &c., that t8 to say, ~t'==M–l.
If a surfaceof the formaM+ bN touch P, the point of contact
ia evidently a point on thé locna just considered,and therefore
the number of saeh ani~cea whieh can be drawn to tonch P,is equal to the number of points in which thia loeu curve
meets that !s to say, !ap times the degree of that eurve.
Reasoningthen M in the last article we Me that the condition
403GENERAL THEOM 0F SUBFACEa.
DD2
th&ttwo surfaces M and N ahouMtouch, coatMMthe coeN-c!en~ of Min the degree M(M'*+2m'M'+8M'~or
and in like manner contains the coe<Sc!ents of in the
degree M(M*+2NMt-)-3~-4M.-8M+6). Moutard, Kf~tM~,
VoLxtX.,p.65.We add, in the fonn of examples, a few theorems to whidt
it does not seem worth while to devote a separate art!cte.
Ex. 1. Two sur&eet U, ]~of the degreet m, n intetsect) the tmatbe)fof tMgentt to their em~e of mtetMc6<m whieh are ttho Me]demd tan-
genta of thé CHt surface, h M~(3)n + 2tt 8).The MexioMi tangents at any point on a MM&ceare genestthg Unes
of the polar quadric of thtt point any plane therefore thMogh either
tangent touchée that polar quadrlc. If then we fonn the condition that
the tangent plane to may touch thé polar quadrie of U, which conditioninvolves the aeeond diniMen~tl* of U in the third degree, and the nrat
di<brent!al<of in the second degree, we have the equation of a Mt&eeof the degree (3~ + 2~ 8) which meeta the curve of intersection in the
pointa, the tangente at whieh are !aBetionaI tangente on K
Ex. 2. In the same case to ând the degree of the Mttace generated
by the in1lexional tangents to U at the ee~Otalpoints of the eatre UV.
ThMie got by eMminatiog t!V< between the equations
whieh are m ~z'tp* of thé degrees respectively )t), <<,m t, <? 2, andin .tytw of the degteM 0,0,1, Z. The TMatt ia thetefote of the de~'ef
OMt(3ot-4).
Ex. 8. To Bnd the degree of the d~etepaMe whieh touches si tm&ce
along its htemeetîon with its HeNiaa. The tangent planes et two eon-
M<Mt!~epoints on thé pttMbolieearve, htttMeet in an InN~doMi tangent(Art. 238)) Md, by thé hot example, tince <f4(m-3), the degree of
the sxr&ce generated by thMe Mextona! tangenta in 4m(m !t)(8~ 4).But eince <ttevery point of thé p<Mb&l!ecurve the two iNBeximMltangenteeo!ncHe, and therefoie the ttM&eeagenemted by each of theM tangentscoincide, thé number just &oad muet be divided by two, and the degree
KquSKdMSM(M-2)(3M-l).
Ex. 4. To Snd the eh<tMetem<iM,M et pt 289, of the devolopablewhich
touches a Mr&ce along any phne section of a Mo'&Lcewhoae d~xee Bt.
The section of the developable by the given ptane h the section of the
gtven Mr&ee, together with the tangents at ita 3))t(M 2) pointa of
innexion. Benee we OMitytmd
CONTACTOP UNES WITH SURFACES.4M
E<. 6. To &td the chMacterMet of the devolopable whieh touchée &
MT&ceof the depee m along ita inteKeetion with a eurfhee of degree t).~tM. f=)M)t(<M-l), a~O, f=m<t(3tM+<t-6) whenoe the other
~n~tatitiM Me fmmd M at p. 239.
Ex. 9. To and the ehMMtwMtÏM of the developable toncMng two
pTen tur&CM,neither of which hea multiple tinM.~<M. "-Mt)(M-ty(M-t)'} e~O, f"=OMt(~-l)(M-l)(M+M-Z).
Ex. 7. To ûnd the charaetetMtiMof thé eurve of intersection of two
deveIopaMet.`
Thé tor&ee< are of degreea and t~, and since each hM a nodal
and cuspidal curve of degrees respectively and m, x' and M', thetefbre
thé <Mrve af inteKeetten hM t~: and n~' + ~M actual nodal and
cuspidal points. The cone therefore which stands on the Mrw and
who<e vettet is My point, hM nodal and empidat edgM in addition to
thoae considered at p. 2SO) and the formuhe there given muet then be
modi&ed. We have M thete 'v') but thé degree of thé KcipMcalof this cone is
Ex. 8. To6ndthe chaMctNMtiesof the developablegeneratedby elinemeetingtwogivencurves. ThbMthe tedproctdof thetMtexemple.Wehavethere&re= tv', ==t'm'm~, <<*= +~'f +Stv'.
CMn'ACT0F MNE8WtTBSURFACES.
471. We now return to the cl&aaof problems proposedinArt. 241,v!z.,to Endthé degreeof the curve traced on a sorEMe
by the points of contactof a !inewhichMttsSesthree conditions.Thé cases we aha!lconmderare: (~) to find the cnrve traced
by thé points of contact of lines which meet in four con-sécutive pointa; (.B)when a !ine la an innexional tangent atone point and an ordinary tangent at another, to find the
degreeof the curveformed by the formerpoints; and (C) that
of the earve formed by thé latter; (D) to find the curvetraced by the points of contact of triple tangent lines. Tothese may be added: (<t)to find thé degree of the surfaceformed by the Unes.à; (b) to find the degree of that &nned
by the Mnesconsideredin (~ and (C); (c)to find the degreeof that gencrated by the triple tangents.
405CONTACT OP LINES WtTH SUttFACEa.
Nowto commencewith problem A; if a line meet a surfacein four consécutivepoints we muetat the point of contactnot
only have P'~0, but atso dP'=0, A'='0, A'P*=0. The
tangent limemuât then be commonto the surfacesdenotedbythe last three equations. We findthe conditionthat this maybe possibleby the method by which the points of innexion,and of contact of double tangents, are determined; B~Xe~~ne C'M<~es,pp. 77, 86.
472. Let three surfaces U, F, W contain <ctoy<!in the
degrees respectivelyX, X"; and a:ye'<e'in degrees/t,and let the points of intersectionof theBesurfaces ailcoincidewith a~'M': then it is requiredto find what furthercondition must be fuintled in order that they may have aline in common. When this is the case any arbitrary plane
<M!+~+<se+~<omust be certain to have a point in common
with the three surfaces (namely the point where it is met bythé common Une), and therefore the result of eliminationbetween P, V, W, and thé arbitrary plane mnst vanish. This
result is of the degreeXX' in aM, and X'+\+X\in !c'y'<o*. But since thé resultant Mobtainedby mnttipiyingtogether the result of substitating in <M!+&~+M+<?!p,théco-ordinates of each of thé points of intemectionof UVrP,this result must be of thé form
Now the condition aw'+~'+ce'-K~M's'O, merely indicatesthat the arbitrary plane passes through !B'o', in which caseit passesthrough a point commonto the three aur&ceswhether
they have a commontme or not. The conditiontherefore that
thay ahould hâve a common !me is n'=0; and this must beof the degree
Hence, by the formulajust given, n !s of thé degree (Ita 34).2~ jMM(<9o~ cot~acttben of lines wMchmeet the surface infour consécutivepoints: or (as we may etdt them) of ~«~
CONTACT0F MNM WtTH SURFACES.406
M~M~M~M~~CM~eM~MM~MC~~MM~~MtM~O a
<!efM fMt~Me<Sof the <~ee llK-34*
473. The equationof thé surface generated by the doubleinflexional tangents is got by eliminating a!'y'!9'!c'between
P'=0, A!7'==0, A'CT'==0,A'P'='0; which resolt, by the
ordinary mie, 19of the degree
Now this resultexpressesthe locnaof points whoseBrat,second,and third polars intersecton the surface and smce if a pointbe anywhereon the surface, tts nret, second, and third polarsintersect in six points on thé sar&ce, we m&r that the resoltof elimination must be of the form !7*3fc<0. The degreeof ~f is therefbre »-
474. We can in like manner aolve problem B. For the
point of contact of an inflexional tangent we have P"'e=0,
A!7''=0, A'~=0: and if it touch the surface again, we have
beaittea ~=0, where W M the dMcrimmant of the equationof thé degree n 8 in \t, which remains when the nrat
three tenna vameh of the equation, p. 187. For W then wo
have X"<=(K-t3)(M-~), /==(K-8)(M-4); and having, as
gave this theorem m 1849 (<~M<Ma~a)~ .&«?? Jb«nM~VoL ïv.,
p. 200). t obtamed the equation in an inconveaient form (~Met~r~JOttnMt VoL p. 3M) and in one more convenient (FAt'&M~Majt
jRwtMt~MM, MM,p. 229) which 1 ahall presently give. But t snbstitate
for my own investigation the Yery beautiful pièce of aaft!ysMby which
PtoCMMtCMmch performed the eBminttion indicated in the text, <~e&,Vêt.ï.T!a., p. C3. As the catcutation is long, and the method, which is
applicable to other ptoMema a!M, deservet to be atudied, 1 have thoughtit better to place it by itsdf in an appendix than te introdMe it hère.
Mr. Cayleybas ohMtvedthat eMetty in the same manner a~ the equationof the Hessian M the tNMfonnaHon of the equation f<-<~ whieh b
eathned for every point of a developable, M the eqMtMn a 0 Mthe
transformation of the equation (p. 330) which is aatMed for every
point on a tuled <ut&ee.
407CONTACT OF UNES WtTH SURFACES.
Thé degree then of thé surface whieh paMes through the
points .B M (a 4) (8M* + 5K 24).
Thé eqïUttMm of thé sat~ace genemted by thé Enee (~)
~A~M~mMMph~iM~~m~a~m)m~~M&M~
tangenta ia jbond by eUminating o!'ye'<p' between thé four
equations <7*=0, A<7'-0, A'!7'-0, tF'=0; and from what
haa been just stated ae to thé degfee of thé vanaNes in each
of theae équations thé degree of thé MMUant !s
Bat it appears,as in the last article,that this fesaltamteontMnsas a factor, U in the power 8(M+8)(M-4). Dividing out
this factor the degree of the su)&ce (b) remains
470. In order that a tangent at the point tc~'to' mayeleewherebe an m&exionaltangent, we must hâve Ai7''=0,(an equation for whieh ~=1, ~–M-1), and betadeswe musthave satisfied thé system of two conditionsthat the equationof thé degree M–22 in X:/t, which remains when the firet
two terms vanish of the equation, p. 187, may have three
foota all equal to each other. If thcBLX', he thé
degrees in which the variables enter into these two conditions,the order of the surface which paaaeathrough the points (<7)
is, byArt. 472, +X.t' + (m 2)\'X". But (see Appendixon thé order of systems of equationt)
The locus of the points of contact of triple tangent lines
Minveat!gatedin !iko manner, except that for thé conditions
CMtTACTOf U!tE8VïtH 8CBPACZ8.408
that the equation jtMt coneideredshould have three Mots alt
equal,we sNb:t!tate'the conditionsthat the same equationehoaldhave two dMtmctpMNof equal roota. It will l~eproved.in the
Appendix that for thMsystemof conditionswehave
The orAerof the surface which determines the points (D)M, theK&M, ~a-2) (M-4) (M-5) (?'+&?+12).
To Sntt the surface generated by the triple tangents weare to eliminatea:ye'M' between !7*=0, A!y'!=0, and the two
conditions, the order of the result being
but Bancethis result containstts a factor D~ in order to fmdthe order of the surface (<7)we are to snbstract MX' from thenumber just written. Snbstttctmg the values firat given for
X' \V and for (M 2) (K a) (? 4) (? 6),we get for the order of the auxface(c),
a number whichprobablyought to be dividedby three.
476.TheM teatamstobeeoBaidered anotherdMa of
problems, viz., the détermination of thé number of tangentswhich a&t!S~four conditions. The foUowmgis an enumera-tion of thèse probleme. To determine (a~the number of
lines which meet in five conaecnûvepo!nta; (~3)the number
ofpomtsatwMchh~thaiB&emoMltaBgentameettn&mrconnseantivepointe; ('y)the number of lines which are doublyinËexIonal tangents m one place, and ordinary tangents m
another; (S)of line inûexîonal in two places; (<)in&exMMtd.
in one place and ordinary tangents in two othem; (~ <~
which toudt in four places. None of these proMemshas as
yet been aatveA: but we can fmd equations which determine
a major Umitto the nmnberof points et,&o.
If a line meet in five consecatrM pointa it touches the
mtrEMeS (Art. 472), since both at the &'st and second of
thèse pointa it is possible to draw a line meeting thé sm&ce
CONTACTOF PLANES WÏTB aMFACES. 409
in four consecutivepoints. The points wthon are pomtaonthe curve PN, mch that the tangent to that cnrve is one ofthé m&exioaal tangents of U. There&re, by Ex. 1, p. 408,thèse points lie on a denved surface wh<Medegree ia
Bat the pomta~3ako lie on the same surface; for these are
evidently donblepoints on the cnrve C~ that ÎBto say,pointsat whieh F «nd jS touch each other. At thèse points atsotherefore the tangent plane to <8'paaaeathrough an inflexional
tangent of U We get then an equation
where is a numerical multiplier, which I believe to be =3,but which possibly may be greater. Another limit ta thenumber of pointsa and 13 is obtained from Pro&ssorC!ebsch*8calculation in thé appendix.
In like manner thé pointa N)y are both included in théIntemectiom of the surfaces U, S, and that foundas the locusof points j&)Art. 474. And other eqaatiom of connexionare
foundin like manner,but not saoMientto determinethé number
of pointa.
CONTACTOF PLANESWÏTH8CBFACES.
477. We can discnas the cases of planes which toach a
NM&ce,in the same manner as we have donethoseof touchinglines. Every plane which touches a mr&ce meets it in aeection having a double point: bnt since the equation of a
plane mcIadiMthree constante, a determinatenumber of tan-
gent planes can be found which wiU &?! two additionalconditions. And if but one additional conditionbe given, anmnnite seriesof tangent planes can be foundwhichwill satisfytt, those planes enveloping a developable,and their points ofcontact tracing ont a curve on the snr&ce. It may be re-
qnired either to determine the number of aointiomwhen threeconditions are given, or to détermine thé nature of the corvéeand developables just mentioned, when two conditions are
given. Of the latter ctaas of problemawe aha!l considerbut
CONTACT 0F PLANES WITH SURFACES.410
two, vis., the diacueMonof the caae when the plane meetathe surfacein a section having a c<Mp;or, when it meets !n
a section having two doaMepomts. Other casée have beenconeida~d by anticipationin the laat aectioa, as for example,the case when a plaae meets in a section having a douMe
point, one of the tangents at which meets in four consecutive
points.
478. Let the co-ordimateaof three points be aiy&V,
:B'ye"«)", a~zto; then thoseof any point on the plane throughthe points wiUhe ~o'+/M)"+tw, ~'+~<y, &c.: and if
we sabstitute these values for ayzM in the equation of the
surface, we shallhave the relation whichmust be satisfiedfor
every point wherethis planemeets the scur&ce. Let the result
ofsnbstitutionbe [ÏT) =0, then [U] maybe written
The plane will touch the surface if the discnmm&ntof thia
equation in X, f vanish. If we suppose two of the pointsnxed and the third to be variable, then this discriminantwill
represent all the tangent planea to the surface which can be
drawn through the Nnejoining the two 6xed pointa.We sh&tlsupposethe point a/y'~M'to be on thé snr&ce,
and the point a!'y's"M"to be taken anywhere on thé tangentplane at that point: then we shall have P''='0, A,,P'*=0,and the discriminantw!!I become divisible by the square of
AP*. For of the tangent planes, which can be drawn to asurface through any tangent line to that sar&ce, two wlll
coincide with the tangent plane at the point of contact of
that Une. If the tangent plane at a~'M' be a doaMetan-
gent plane, then the discriminantwe are considering,insteadof being, as in ether cases, only divisible by the square of
thé equation of the tangent plane, will contain its cubeas a
CONTACT0F H~NES WMHSURFACES. 41t
factor. In order to examine thé conditionthat this may be
eo, lot us for brevity write the equation [U] M fbUow~thécoeSdonta of X* being sapp<Modto vaniah,
T tepresenta the tangent plane at the point weare oonsidering,C !t8polar quadric,wh!ie =:0 le the condition that a/y'e'Vshould lie on that-polar quadric. Now it will be found thatthe dîa<anminantof [P~j Mof the fmm
where is the disaficfunfmtwhen T vanishes as well as )yand A, In order that the discriminantmay be divisible
by T', some one of the 6tctomwhichmultiply 2" muâteithervanishor be divisibleby
479. Fîrst then let vanish. This only denotesthat the
point a/y~'to" lies on the polar quadrieof .cye'M' or, sincoit aiso lies in the tangent plane, that the point a;"y"e"w)"lieson one of the inSexionattangents at a~'e'M'. Thns we teamthat if the class of a Mrface be p, then of the p tangentplanes which can be drawn through an ordmary tangent line,two coincide with the tangent plane at its point of contact,and there can be drawnp 2 distinct &om that plane: butthat if the line be an inflexionaltangent, three will coincidewith that tangent plane, and there can be drawn onlyp- 8distinct from it. If we suppose that a:"y'V<c"bas net been
taken on an in6exionaltangent, willnot vanish,and wemaysetthis factor asideasirrelevant to thepresentdiscaeston.
We may examine at the same time the conditionsthat T
shouldbe a factor inJS'LC', and in
The problem whicharises in both theaecases is thé follow-
ing Suppose that we are given a &mction whosedegrecsin <cy~'M'in a!'y<o", and in a~c are respectively(~,/<,/t).Sappoae that this représenta a surfacehaving as a multipleline of the order the une joining the first two points; or,in other words, that it represents a series of planesthronghthat tine to &id the conditionthat one of these planesahould
CONTACT 0F FLANES W!Ta SURFACES.412
be the tangent plane T whose degrees are (<t-1,0,1). If M
any arbitrary line wMth meeta T wm meet P, and thereforeif we eliminate between the equations y==0, F=0, and the
equations of an arbitrary tine
the résultant B mnat vanish. This Mof the degree in <tM,in <t'&'o'<f,and in ai'y'e'V, and of thé degree ~(M-1)+X.in ai'tc'. But evidently if the assumed right Une met theUnejoining a:y~'M',a~"y"M", wouldvanish even though Twere not a factor in V. The condition (3f==0), that the twounes shouldmeet is nf thé crat degree in all thé quantitieswe are considering and we aee now that B is of the formJtfJB'B* remainBa fonction of iBy-s'tc'alone, and is of the
degree ~(<t-8)-t-
480. To apply this to the case we are constdenng,a!ncethe dMcnmoMmtof [U] KpMMntaa series of pitmes throughey~tc', a/y'W, it followathat ~C aad both repreBentplanes through the aame line. Tho nmt is of the degrees{a(K-2),2,2}, while is of the degrees (K-~)(tt'-6),M'-SM'+K- 6, n'-SM'+M-e, as appears by suhtractmg the6nmof the degKesof f, and (B*- J.C)' fromthe degreesof the discriminantof [!7], which is of the degree M(o–l)*in all the variables. It Mtows then from the last article thatthé condition (R=0) that T ehouldbe a factor in ~C
is of the degree 4 (M-8), and the condition (jK'=0) that Tshouldbe a factorin is of the degree (n- 2) ()t* M*+ M-18).At all points then of the intersection of C~and If the tan-
gent plane must be considered double. H is no other thanthe Hessiam; the tangent plane at every point of thé carveCEfmeets the surfacein a sectionhaving a cusp~and ia to becounted as double (Art. 888). The curve CK' !a the locus of
points of contact of planes which touch the surface in twodistinct points.
481. Let us conalder next the series of tangent planeswhiohtouch along the eurve !7K They form a developablewhose degree :s p=a2n(M-8)(3a-4), Ex. S, p. 403. Thé
CONTACTOF PLANES WTTB SUBFACRS. 418
dass of thé samodevelopable,or the number of planes of thé
System which can be drawn throngh an MNgned point, Hf e=4M(<t -1) (a 2). For the points of contactare evidentlythé intersections of the curve EETwith thé firat polar of the
assigned point. We can aieo detemine the number of
at&tMntoyplanes of the System. If thé equation of P, thé
planea being thé tangent plane at any point on the curve PB,be <+~'+M,+&o.=0, it tB easy to ahow that the direction
of the tangent to DB~:8 in the line'=0.
Now the tan-
gent planes to F are thé same at two consécutivepointaproceedmg along thé inSexional tangent y. If then «, donotcontain any term i~, (that is to say, if the Mexional tan-
gent meet the aurface in four coMecntivepoints) the directionof the tangent to the curva KS is the aameae that of theinnexMmattangent: and the tangent planes at two coneecndve
pointa on thé carve PT will be the same. The number of
Btationarytangent planes is then equal to the number ofintersectionsof the curve UH with thé snf&ce& But sincethé carve touchesthe surface,as wiUbe shewnin the appendix,we have c[!='3<!(H-a)(llM-24). From thelle data all the
siNgnIarMesof the developablewhich.touchesalong SB canbe determined,as at p. 2S7. We have
Thé developablehere consideredanswersto a cnspidal Uneon thé reciprocalaar&ce,whose a!ngal<mt!e8are got by inter-
diaBgmg/t &nd etand jS,&c.in the above&)rmche.The dam of the developabletouching along !XK,whieh M
the degree of a doaMecurve on the reciprocalsurface,is t)eeaas above to be !t()t-l)(K-2)(K'–M'+M-12). Its other
amgttiMttteawill be obtained in the next section,whcre weshanabo detemine the numberof solutionsin somecaseswherea tangent planeMrequiredto fuMItwo othorconditions.
THEOBT0F MCÏMOCAÏ,SURFACES.414
THEORT0F KECïfMCAl,SURFACES.
482. Pnderstandïng by the ordinary smgutanttes of a
surface,those which in general cxMteither on tho surface or
ite reciprocal,we may make the following enumeration of
them. A sm'&cemay have a douMecurve of degree b anda cuapidalof degree c. The tangent cone determinedae in
Art. 846, includesdoubly'thé curve standing on tho double
curve, and trebly that standingon the cuspidal curve,M thatif thé degreeof the tangent coneproper be a, wehave
The clam of the cone <tis the saine as the degree of the
reciprocal, Let a have 8 double and « cuapidaledges. Letb have k apparent double points, and t triple points which
are abo triple pointaon the sar&ce; and let c haveAapparentdouble points. Let the curves 6 and c intereect in <ypointa,which are stationary pointa on the former, in j8 which are
stationary point: on the latter, and in < which are Bmgohtrpointa on neither. Let the curve of contact a meet b in ppoints,and c in <ypointa. Let the same letters acoenteddenote
singatantiesof the reciprocalsurface.
488. We saw (Art. 247) that the points where the curveof contactmeeteA'<7give rise to caspidal edges on the tan-
gent cône. But when thé line of contact consista of the
complex eurve a+3&+3c, and when we want to determinethe numberof caspidat edges on the cone a, the points whereb and c meet A'!7 are plamiy irrelevant to the question.Neither shaU we hâve cuapidal edges answering to aU the
points wherea meets A'O, since a commonedge of the cônesa and c is to he regarded as a cuspidal edge of thé compkxcone, although not so on either cone conaideredeeparately.The following&rmnlœcontain an analysteof the intersectionsof eaohof the curvesa, b, o, with the surface A'0,
THEORT OF RECÏPROCAÎ.SURFACES. 415
The ret~er can BMwithoat dinicnlty that the pointa tNd!<*atedin these &rmu!Mare included in the intersections of A'!7with <t, e, respectively: but it !s not M eMy to aee thereason for the numerical multiplierawhich are naed in thefbrmulœ. Although it u probably not impossibleto accoantfor these constants by otpriori reasonmg,1 prefer to explainthe methodby which 1 was led to them indacttvely.*
484t. We know that thé reciprocalof a cubic ia &aurfacoof the twelfth degree whichhas a cuspidaledge of the twenty-fourth degree, since ita equation is of the form 64~y*,where jS Mof the fourth, and T of the mxth degree (p.876).Each of the twenty-seven lines on the surfaceansweK to adoaMe line on the reeiprocal (p. 878). The proper tangentcône, being the reciprocal of a plane section of the cubic,is of the mxth degree, and bas nine cuspidal edges. Thus wehave <t'=6, y.=a7, c'=24, M'==12,<t'+8&'+8<=!2.11. Theintersectionsof the corvesc*and &'with the line of contact ofa cône a' through any assumedpoint, anewerto tangent planesto the original cubie, whose points of contact are the inter-sections of an aseomed plane with the paraboHccurve KB,and with the twenty-aeven lines. ConseqaenÛy there are
twelve pointa o' and twenty-seven points one of thelatter points lying on each of thé limesof which the nodal
t!ne of thé reciprocal mr&ce is made up.Now the sixty points of intersectionof the curve d with
the second polar whîch is of the tenth degree, comist ofthe nine points <c',thé twenty-seven points and the twelve
points < It is maoifest then that thé last pointa must
The Smt attempt to exphin the e&et of nodal aad ccapidat lineeon the degree of the teciprocd anr&ee, ~M made in the yeM 1M7 in
two papera which 1 contributed to the ChotM%w«o~ DttZ~MtJM<t<~«M<t<M<!<
Jb~~fH< Vct. n., p. M, and iv., p. t88. It was not tiU the close of thé
year 1849, howeter, that the <tiMMery of the twenty-Mten right lines
on a eaMe, by enabling me to ferm a clear conception of the nature of
the reciprocal of a cubic, kd me to the theory in the foMnhere exphoned.Some few additional detaila will be found in a memoir whioh1 eomtnhnted
tothe 2h!<Mo<!fM)«~&'y<MA~ea<f<N~,Vo!.xxin.,p.46t.
THEOBY<? RECtFMCALSURPACtS.416
countdouMe,sincowe cannot satisfy an equation of the form
S<t+27&+12c'=60, by any integer vahea of«, c except
1, 1,3. Thus we are led to the arst of the équations(jl).Conmder~now the points where any of the twenty-seven
Mnesmeets the same sortace of the tenth order. The points~3'anewer to thé points where the twenty-seven right Unestoach the pat&hoUccurve; and there are two mch points oneach of thèse Unes (Art. 286). There are aleo five points <on each of these Hnes (Art. AS5),and we have just seen that
theMisone point p. Now since the equation a +2&+5<!e=10,can have only the systems of integer solutions (1, 2, 1) or
(8, 1, 1), the ten points of intersection of one of the lineswith thé secondpolar muet be made up either p'+~3'+< or
3p'-t-+< and the latter form is manifestly to be rejected.But consideringthe curve b' as made up of the twenty'Mven
lines, the points<'occur each on three of thèse lines: we arethen led to the formula &'(K-2)~+2p'+3('.
The example we are considering does not enable us todeterminethe coefficientof -yin the secondformula becanaethereare no pointsy on the reciprocalof a cubic.
Lastty,the twohundred and forty pointa in whichthe cnrvec meetsthe secondpolar are made np of the twelvepoints <r',and the nfty-&nr pointsj8'. Nowthe equation 12a+ 64&=840
onlyadmitsof the Systemsof integer solutions(11,2), or (2,4),and the latter is mani&f)tlyto be preferred. In this way weare led to asaign atl the coemcientsof the équations (~) ex-
cept those of
485. Let us now examine in the same way the reciprocatof a sarface of the M**order, whieh bas no multiple points.Wehave then M'=M(?-!)', M'-2=(M-2)(M'+1), a'=M(M-l)and for thé nodal and cuspida!curves wehave (Art.256)
Thé number of cuspidal edges on the tangent cone to the
reciprocal,answering to the number of points of itimexîonon
a plane :ect!on of the original, gives us <'=8M(a–2). The
points p' and.<r',answer to the points of intersection of an
Msumedplane witb the curves C~ and CET (Art. 480)
417THEOM 0F RECIMiOCAL8UMACM.
hencep'= M(M 2) (~' M*+M-12), <r*=4t (x 3). Sahstitutethese values in the formula <t'(n'-2)=<t'+~'+8<r', and it iaaatîsSedtdentîcalty, thus verifying the nmt of formntœ (~).
We shall next apply to thé samo case the third of the&tcmhe (~.). It was proved (Art. 481) that the number of
points ~3'ia 2n(n-2) (llK– 94). Now the intersectionsof the
nodal and cuspidal carves on the roc!procalsnr&ceanswer tothe planeswhich touch at the points of meeting of the cnrvea
UH, and D~ on the original surface. If a plane meet thesur&ce in a sectionhaving an ordinary doublepointand a cnsp,smee fromthe mere fact of its touching at the latter point it !a
a double tangent plane, it betonga in two ways to thé systemwhich touches along H5'; or, in other words, it is &stationary
plane of that system. And sinee evidently the points ~3'areto he includedm the intersections of the nodal and cuspidalcnrve, the pointa P, .B, iT must either answer to points ~S'or points y'. Assuming, as it is naturat to do, that the
points ~8count double among the intersections of CBK~we have
But !fwe snbstitnto the values already foundfor e',M',or',j8',the quantity c'(M'-a)-2<r'-4~8' becomes alao equal to thevalue just MStgnedfor y. Thua the third of the &rmu]œia ven&ed. It would have been sufficientto assame that the
pomta coant times among the intersectionsof CEû~ andto havewritten the third ofthe fonnubaprovidoMÏly
when, pMceedmgas above, it would bave beenfound that the&H-nMtIecouldnot be satisfiednnless = 1, -=ia.
It only remains to examine the secondof the &nnnlœ (~).We haTejiMtassignedthé values of all the quantities involvedin it except(. Sabstitntmg then thesevalues,we&td that thenumber of triple tangent planes to a em&ceof the M**degreeis given by the formula
THMRT <M*NBOn'BOOAt SUN'ACES.418
486. It wu proved (Art. 248) that the pointa of contactof those edgesof the tangent cone which touch in two distinct
pointa lie on a certain surface of the degree (K-2)(n–8).Now when the tangent cone la, M beforo, a complu cône
a-~2&+3e, it is evident that among theae double tangentawill be ineludedthose commonedges of the coneaab, vh!ch
meet the cnrvMa, b in distinct points: and tunNarly for theother pMMof cones. If then we denote hy [o~]the numberof thé apparent intersectionsof thé curves a and that Nto
say, the number of points in whieh thèse curves aeen from
any point of space seem to intersect, though they do not
acttMJlydo M; thé followingformulea will contain an ana-
tymaof the mtemectionsof < e, with the surface of the
degfee (« 3) (M 8)
Now the number of apparent intersections of two curves is at
once deducedfrom that of their actual mtemect!ons. For if
conesbe deacribedhaving a commonvertex and standing onthe two carvea, their common edgea mut anawer either to
apparent or actnal intetsectiona. Hence,
The fimt and thîtd of these equations are MtMûd td~tteaNyif we subatitute for y, &c. the values need in thé laet
If the <ot&aehave t nodal earM, but no caapidat, there wiU etm
~e a determinate number < ef cuapidal po!n<< on the nodal eurve, andthé above equation Kee~M thé tnodjHMtdon [a6] = <tt 2~)-1. In de-
tenaMn~ however the degree of thé reoiproeal surface the qMndty [ab]i< eiiBdMtedt
BB2
article, to which wo are to add ~=a(M-2)(M'-9), <'='0,and the value of A given (Art.481),vis.
Thé secondequationenableaus to determinek by the equation
&om this exprestMnthe rank of the developableof whichb Mthe cuspidaledge canbe ca1calatedby the formula
Pottmg in the vaines aheady ohtamed for thèse qumtt~eo,we 6nd
This is then the rank of the developable formod by thé planeawhich have double contact with the given Bar&ce.*
~87. From &trmalfe and B wo can catcnlate the diminu-
tion in the degree of the reciprocal caused by the singularitieson thé original tmr&ce enumerated Art. 482. If the degree of
a cone diminish &om m to M– tbat of îts reciprocal diminithes
from Mt(m–l) to (m-~)(M- 1); that is to say, is reduced
by ~(2Mt–l). Now the tangent cone to a surface is in
general of the degree «(M–l), and we have seen that when
the surface haa nodal and euspidal lines this degree ia reduced
In order to vet!~ the theory it would be MeeMMyto show that thianumber Jï eoMctdea with what may be deduced from Ex. 6, p. 4M. Intha &Kt place the developable generated by the ouspidal ourve on the
reciprocal mrface corresponda with that whieh envelopes the given anr&ee
a!<m{[!7B; and which, by the eMmpte cited, ought to be of the degree28(tt !!)*,but if we eabtMCt&om thia the number ~9,we gat the ~atne
already determined. In like tMMteif, if we take the Mt&ce envetopingthé given Mr&ee along !7K' (Art. 480) and NtbtfMt from the degreedetermined, M in the example cited, <'y+ j9+ <M,we get not J! but ~B.Poaeibly thia may be becanee all the tangent planes which envelope the
developable in que$tmn are doab!e tangent ptanet} but it muet be ownedthat there are pointa in all this theory which need ftather explanation.
TBEOM 0F BECÏPBOCAÏ.aUBPAOES.420
bya&+9«. TheMia&comeqnentdimumdoninthed~Mtofthereciptocdaarface
Bnt the exigence of nodal and cuspidal curves on the nar&ct
causeaa!aoa diminutionin the nTtmberof dombleand cuapidaledgesin the tangent cone. From the diminutionin the degreeof the reciprocalsurface jtMt given must be sabtrMted twietthe (MminnUonof the number of double edgea and three ~mefthat of the cuspidaledgea. Now from &)nnahaA, we have
Bat since if the aat&ce had no multiple linee the number of
ca:pt<Medgesonthe tamgentccmewouldbe (o+2&+ 8c)(M-2)tthe diminutionofthe number ofcaspidal edgeais
The tt!mnmtM)ithen in the nmnber of donHeedgea ia given
by the form.
488. The tommbe and B can be thrown intoa formmore
cmtveniemt&rTMe.-IfwetMnembert!mtet+St{+8c'a)t(<t-ï),the &rstof&nmbe B may be written
THEOMTOP MNKMCAL SURFACES. 421
But <«-33-8<f is ?' the degreeof the tedpmcaï aor&ce.Henoe
Thé tntthofiHa equation maybeotherwiaeaeen&omthcconeiderstionthat a, the cnrve of simple contact from say one
point, intersecta the Ërot polar of any other point, either inthe <t*pointa of contact of tangent planéepaee!ngthrough the
Mnejoining the two points, or dse in the p points where a
meeta or thé o' pointa where it meete et, einceevery nKt
polar paasea throagh the carvea t, c.
Adding the eeoondof &nna]<eB to four tumeathe secondof formule and giving JBthe Mmemeaninga in Art. 486,weget, in like manner,
an equation of which 1 donot Me the geometricalexphmattonatttonghevidentty the B pointa on 6 the tangents atwhich
meet any line are inclnded among the p points on b which
are pointe of contact of tangent planes through that line.
If the last of each of thé <bmm!<abe treated in like NMamer,andif wecall ~8'the order of the devetopaNegenofated by the
carvec; that M,if wewrite
489. The efbet of multiple lines in diminishingthe degreeof the McîpMcatmay be otherwieeinvestigated. The pointsof contact of tangent planes which can be drawn through a
given line are the intersections with the surface of the curveof degree (M-1)' which is the mteraectionof the Btst polaraof any two points on the line. Now let us fnt considerthe
case when the surface has only an ordinary doublecurve of
degreeb. The first polars of the two ponitspam eachthroughthis curve, so that their intersectionbreaks up into this curve
b and a complementalcnrved. Now in looking for the pointsof contact of tangent planes through the given line, in the
nmt place, instead of taking thé points where the eomplexcarve ~+J meets the surface, we are only to take thoae in
MVEMPABMB SURFACES.4M
whieh d meete It, which caquesa réductionbn !n the degreeof the reciprocal. Bat, forther, we are not to take atl the
pointa imwhich meeta thé surface: those in which it meeta
thé corve & being to be rejected; those being in number
2b(n 2) r (Art. Xll) where r is the rank of the system &.Now thèse points conaist of the points on the carve b,the tangents at which meet the line through which we are
eeeMng to draw tangent planes to thé given surface, amd of
S&(M–2)-2f points at which the two polar urfaces touch.These last are cuspidal points on the double curve b; that isto say, points at which the two tangent planes coincide,and
they count for three in the intersections of thé curve d with
the given surface,since the three surfaces toach at these points;while the r points being ordinary points on the doaMe line
only count for two. The total reduction then is
wMchagrees with the precedmg theory.If the carve b, instead of being merelya double curve,were
a multiple cnrve on the aar&tceof the order p of multiplicity.1 have found for the redactMn of the degree of the redprocal(aeefR'<!M<!C<tMMo~t~JB<'ya?A'M~~ca<&Mty,Vol.xxnt., p. 485)
for the reduction in the number of cuspidaledgea of thé coneof simplecont&ct
MEVEMPABMSCBFACE8.
490. The theoty just expttumed ought to enable tm to
accoamtfbt'~e&ct~tthedegreeoftlM) MCtproodofa a
Themethodof thiaMUdeia not apptiedto thecasewheMthe surfaceha a cuspidalearve in the Memoir&o)mwhichï <!te,and 1havenot now
leimtteto attemptto repairthe onnMt<m.
MVEUM'ABLE80BFACM. 4S8
developablereduces to nothing. This applicationof the theoryboth verifiesthe theory itsdf and enablesus ta determineMma
eingatarMes of developablesnet given, p. 289. We use thenotation of the section referred to. Thé tangent cône to a
developablecoMÎstsof n planes; it has therefore no cuspidaledgesand ~a (n -1) double edgea, The dmple line of contact
(a) conmstaof Mtinea of the ~Btem each of which meete the
cuspidal edge m once, and the double Unea; in (r-4) points.The lines Mtand a?interaect at the <[points of contact of the
stationaïy planes of the system; for since there three con-secutive Hnes of the system are in the Mme p!ane, the inter-Mctton of the &rst and third givesa point on the line a~*
We have then the following table. The letters on the IeA-hand side of the equationsrder to the notation of this Chapterand thoaeon thé right to that of ChapterXI.
M=y,<!==M,6=iB,C=M;~=H(f–4), 0'==M,<=0, ~S=~3,X=~,~a;
and the quanttties<, 'y) femam to be determined. On sub-
atttutimgthese values in &mtcJœ and .B, pp. A1A,418, we
get the system of equations
The UrBtof theae equaiiom is identicallytrae, and the fourthis BtttMËedby the help of the equation,provedp. 286,
If we eHmmata<ybetween the third and Nxth equations, we
obtainàbo an equation aheady provedto be tme (Chapterxi.).The three remaining equations determinethe three quantities,whoaevalues have not before been given, vis. t the namberof "pointa on three Unes" of the aystem; 'y the number of
It i< onlyonMeoantoftheirocourrencein thh eïMnptethat1wasM to inoludethepointein thetheoty.
DBVEMPABUS SOKPACE8.424
pointe of the system through each of which pttmManother
aoN-conMcat!velino of the syatem; «Bdk the ntMabefof ap-
parent doaMe pointa of thé nodal line of thé developable.These quantities being determined,we camby an interchangeof letters write down the reciprocal singuJarities, v!z. the
number of planes throngh three ImeSt"&c.From Art. 488 can be deducedJS the rank of the develop-
able of wh!ch<cis the OMptdaIedge. For we have
E]t. 3. To Cnd the <htgalMMe< of the developable gencMted by a Hne
K<&~ twice on a p~en catve. Thé pt<mMof this ey~tem are evidently
planes thro~h two linea" of the original <y<temthe etMa of the ty~temfa therefore yj and the other mngatantie< are thé Meiptocab of thoae ofthe eyttem whoae tutp!dal edge !t w, caleotated in this article. Tha* the
rank of the syotent,or the order of the devetopaUeis given by the formula
491. Sinee the degree of thé reciprocalof a raleAMti~tceredaces aiways to the degree of the original sur&ce (p. 849)thé theory of redprocal snr&ceaought to aocoant for this re-
dacdon. 1 havenot obtained this explan&ttonfor roM Bnr&ces
în gene~ but some particular taees are exammed tmd ao-
counted for in thé Memoir in the 3~MM<M<MMof t&eRoyali~~c<!<&Mya!readycited. IgtveonlyoBeexsmp!ehere.
426MYEMPABLESmOfACES.
Let the equationof the tMt&cebe derivedfromthe eliminationof< betweenthe equationa
where <t,d, &c. are any linear functioasof the co-ordinates.Then if we write &+~=~ the degree of the surface h ~t,having a double line of the order ~(~-t)(~-2), on whichare t(/S)(~-3)(~-4) triple poîiita. For the apparentdouble points of thia doaMecnrve, we have
values which agree with what was proved, Art. 459, viz. thatthe number of cuspidaledgesin the tangent cone is dumaMhed
by 8&(/t-S)-3<; while thé double edges are dimimehedbyS!&(K-8)(M-8)-4A. InveriiyiDg theMptu'ate&mabe Bthe remark, note, p. 418, must be attended to.
492. It may be mentionedhère that the HesManof a ruied
sm'&cemeets thé surface onlym its multipleHaea,and in the
generators each of which ia mtersected by one conaecn~vo.
For, p. 847, if xy be any generator, that part of the équa-tion which is only of the <!mtdegree in mand y is of thé form
(aw+y<c)~. Then, Art. 2S6,the part of thé Hessian whichdoea not contain x and y is ·
which reduces to But a;y mtersecta only in the pointswhere it meets muMple linea Bat if the equation be of théform «!c+< (Art. 2M) the Hessianpameothrough ay. Thus
in the case we have consideredthe numberof Imeawhich meet
one consecutiveare eamiyseen to be 2 (/<-2); and the carveUH whose order is 4~(~-2) consistaof theae Unes each
counting for two and therefore equivalent to 4(~-2) inthé intersection; together with the double line equivalent to
4(~–1) (~-8). Again, if a aurface have a multiple line
NEVBMPABÎ.B SCNFACE9.426
whose degree ia M, anAotder of multiplicity it ~nHbe
ItBeof order 4(~-1) on tho ReMtan, and will be equivalenttoémp(~-t)on&e<M've !7B. New the ruled surl'ace
generated by a line resting on two right lines and on &curvem (wbich is supposedto have no actual multiple point) Moforder 2m, having thé right lines M multiples of order m;having ~m(<?-!)+& double generatots, and 2f genetatorawhich meet a oonsecuti,eone. Comparingthen the order ofthe curve CB with thé Mun of the orders of the carves of
which it is made np, we have
an équationwhichis iden~càUytme.
If we form the Heoaian of thé developablea*M+y~, it
appem in like manner that we get the developable itadf
mntttplledby a series of terma, in whichthe part independent
<a~ï.(~)~TM.prove.th.tthe
~dx" dto~ CcZxdrcJ~·Pro-Hesatan (aeep. 838)meets each generator in the &'stplacewhere that generator meets that is to aay) twice in the
point on the carve M, <mdin f– 4 points on the cnrve a;; andin the second place where the generator meets the Hessian
of «; that is to say, in the Heasisn of the aystem formed
hy thoser 4 points combinedwith the point on m taken three
times in whieh Hessian the latter point will he indu'ied farn*timM. The interaectionthen of the generatorwith the Pro-
nesdan consistaof the point on M taken aix tunes, of ther-4 points on a~and of 2 (f- 5) other points.
( 427 )
APPENDIX I.
ONTBBCAMUMT8OPQVATEBMON8.
1. THE Caîooinsof Quaternionshaving been anccesaMIyemployedby Its invontorSir W. R. Hamilton in the deductionof geometrical theorems, it may seom proper to add someaccoontof it to that which bas been given in the precedingpages of other methods of inveatigating the properties of
space of three dimensions. Neither the apace now at mydisposai, nor my knowledge of the snhject, allow me to
attempt heïe to teach this catcnins; bnt in the followingsketch 1 hope to give the reader some idea what quaternionsare, and how they may be used in geometrical enquiries;re&mmghim for further informationto Sir W. B. Hamiltons
papers "On SymbolicalGeometry" in the C~aK~&~eandDublin JMot<XeNM<M<~J<M<nM?,to his "Lectorea," and to his
ibrthcoming Elémentsof Qa&temiom."Yectors. In AlgebraicGeometrythough the symbolsa), y,
&e. are used each with référence to a line measured in acertain assigned direction,yet in the equationsemployedthèse
eymbo!sdenote merely the NtayM<M<!Mof the linea which theyrepresent; and the eqna~onsonly express that certain arith-metieal operatioMare to be performedon the numberswhioh
express the ratios of each of the Unes ic, y, a to the linear
unit. Thus if we form the snm iB+y+e of three known
lines, thé result ia a line of determinate length but of no
assigneddirection. In the quaternion caicuins a symbol de-
noting a Unemust aiwayaexpressdirection as well aa lengthand if for instancewe form the sam a?+y+~, it is necessaryto assignthe directionaa well as the length of the line whichîa the remdt. In this calcahmthen me signa + and areueed not with re&rence to numerical additionor sahtMctMB,
ON THE CAMUMM 0F QUATEBNION8.428
but with re&Mnce to direction (aa we pt'oceed to expïain),and denote geometrical, not atgebïa!cat, addition and sab-
traction.
2. Let the line or vector .~B be <mdet8toodto denote thé
operation of pMceedmgfrom the point A to the point B;then BO in like manner would denote the operation of pro-
ceedingfromB to <X The stgn + may natar&Uybe employedto denote the coneecutiveperformance of these two operationa;thns AB+J9C would denote that we proceed omt &om
to B, and then from B to C; and since the teaalt ia
the aame as if we had gone direct from A to C, we have
~J?+ BC=*~C~ The somof two vectomthen is the diagonalof the parallelogram of whieh theae Unes are ad)acent ddes.If AB and J?C'were portione of the Mme right line, then
their mm wouldbe the ordinary algebraîo sum of the two
lines; and it is easy to ne by successiveaddition that if a
dénote any vector, and m any anthme~cal moIttpUer,ma
denotes a vectorcoincident in direction with that represented
by a, and in length bearing to it the ratio M 1. Two vectom
are eaid to be equal if one can be moved without rotation
so as to coincidewith the other: that is to say, two eqnatlengths measuredon parallel Unes are said to be equal. Bythe help of thia convention we can interpret and verify thé
equation <t+~=}+<t. Let the vector a be represented bye!ther of thé equal Unes ~E~ J?C,<md& by elther of thé equ&l Unes
DJ~JS'B;then if we take o6Mt
we have o-~t=jdJS, but if we com-
mence with &wo have &+a==.DC';and these rem!ts aroequaiNnce
~Band~CMeeqnalandparaBdL ItN evtdentonm-
terpretation of thé equation that
Thus we see that the ngn + when geometrtcaHyinterpretedaa here proposer,conionas to the ordinary rules of atgebmic
addition, Tiz., to the commutativehw o+&=it+<~ and the
<MMCM<tcelaw (a+~)+o=~+(&+c).
ON THE CAMOMM OP QttATEBNtONS. 43&
8. Denoting, M befora, by AB the operation of goingfrom to -~J? naturally denotesthe revereing of this
operation,v!z., that of goingfromB to BOthat ~JB+JMe'O.It can easily be deduced hence that if <t+~=c, o=e–
Sinoe tbe addition of linesaccordingto the method juet ex-
plained corresponds exactiy to the compositionof mechanical<brcesacting on a point, we can prove,as in Mechanica,that
any lino may be reBolvedinto the smnof three lines whosedireetionaare those of three given rectangniar axes. If no~
nmt lines measured along the axes of a:,y, as tespec~vetybedenoted by <)J< le; and if the munencal ratios which the
lengthsof the co-ordinatesof any pointP bear to the unit linebe denoted, as in algebraic geometry,by <B)y, e, then in this
catcaluBthese co-ord!nateswill be denotedby ix, ks re-
apeetively,and the vectorfromthe originto P will be denoted
by M:+~y+&e. And since any vector Mequal to a panJlelone through the origin, tMre « uo MC<M'<oA«Amay <!0<
<apMMed'<M<&!j~M <0;~+&If a, be any two co-initid vectoMit is easy to eee that
..t" is a vector drawn from the same origin to the point~Mtwherethe !me joining their sxtrem!t!esis eut in the ratio hM,
and that denotea a vector terminating in the
plane tbrough the extremities of <t, -y. If <tand Il be bothof unit length, &[+M~ makes with a and /3 angles whose
ines are in the ratio ?:M. These pnncipteamay be used toestabliah geometrical theorema. Thus ~(a+/3+-y+~) is thevector to the centre of gravity of the tetrahedronfomed bythe extremMesof a, Y,o; from wlùchform inferencesmaybe deduced aa in Ex., p. 6.
4. 6tM<M'KM!M.We have now ehownhow UnescoBmdeMdwith respect to their directionas well ae to their magnitade
may be added and a&btraoted,and we comenext to speakof
multiplicationand diviaion. It M not obvionswhat aenaeweare to attach to the prodnct of two lines, but it is natural
to mteïpMt theqaotîemt
aa denotmgthé operationneeemary
4M ONTBBCAMOUtSOFQOATMERNMNS.
to change thé Une into the line et eomat
~='<t. If the
vectors<tand ~Sbe portiona of the same !me,it !e evident thatthe quotientMa numerical constant,or, as Sir W. B. HamiltoncaHsIt, a aco&N'but, when this Mnot the case, in order to
change~8into a we have aot only suitably t<talter its length,but aiso to tttm it throagh a certain angle in a certain plane.Now we have seen that a vector is redacible to the sam ofthree distinct terme, and we m!ght have foreaeenthia becaueein order to deteraune a vector we muet know three things,viz., ita length, and ita direction-mines, equivalent to twomorecondttîons. But to determinea geometricalquotient four
things are necesaary,viz., the numerical ratio of the lengthsof the two lines compared,thé angle through which one mustbe tumed in order to coincidewith the other,and the direetion-coMneaof the plane of that angle, equivalent to two morecond!tionB. We shall presently showhow to express any snch
quotient as the snm of four irreducible terme: it is thenceca!led a quatemion. It is agreed on that the four elements
just mentioned BhaUbe &M~MeK<to determinemch a quotientas we are considering that in to say, that two quotients are
Mtd to beeqnal ='
Srst, if the lengths of the lines be
proportional, a /3 'y 8; aecondiy,if the angle betweena andbe equal to that between <yand o; and thirdly, if aIl four
limea be parallel to the aame plane. In other words thé
geometrical ratio of two !!nes M consideredunchangcd, not
only if both be mcreasedor duninlahedin the same proportion,but a1so if they be tumed round in their plane their mutualincUnattombeing nnaltered.
& Two geometricalfractionshaving a oommondenominator
are added by adding their numerators: that is to say, we
have ~+'-=– as in common algebra. We can thus00 0
K~< Miyauch&ac~onto one, the two lines in which are at
right angles to each other. For if the fraction be, </ divided
by c, we can reaolve 'y into the sam of two lines <t+~,
ON THK OAMUMJB OP QOATEBNIONS. 4M
one of them in the direction of S (imfact the pto~oetMnofyonS),andtheotherperpentdiculartoit. Nowemoeetis Mpposed to be in the aame direction as c, their ratio is
a mère nnmber or acalar,while thé ratio of 13to c ie that oftwo rectangular Unes. Thus then we can reduce every qoa-ternion to the form f8+ F, the aum of a scalar part and avector part, the latter part being so o&Hedbecause we ahaU
preeently see that thé ratio of two reetangnlar lines can be
adequately representedby a vectorperpendienlarto their plane.A quaternionmay he resolvedin another way via. into a
nomerîcal factor multipliedby the ratio of two equal lines. We
haveoh~ioudy-q. 1=
for if we tiret tum <yinto /3 and"1
then ~3 into a, the rasait is evidently the tuming into eh If
now ~8he sappoaedto be a line equal to 'y, and in the directionof a, the ratio of a to 13is a mere number; and the ratio ato 'y ia resolved into the product of this number into the ratioof the eqmaltineB/3 amd'y. Sir W. R. Hamilton calla thia the
resolution of a quaternion into the produet of a tensor anda fe~M' the tensor being the number express'ng in whatratio the line yie to be mereasedor d!mm!shedin order to bemade equal to ~9,and the versorexpresmig through whatangleit is to be tcmed.
Thus suppose that the symbol J denotes the operation of
tmning a Une round throngh a right angle in a plane per-pendictdar to the vector i: [in order to fix the ideas we may
agrée that the direction of the rotation aha!l be that of théhands of a watch aa we look along t:] then mI denotesthe
operation of tnming the line round aa before,amdat the Bametime altering the length in the ratio 1.
Thus then if the denominatorof a fraction be aime ofunit length, and ita numerator of length if the angle be-
tween them be and the unit vector perpendicular to their
plane be p, wemay first reaolveinto the portionscoe~, sinCmeasured in the directionof the denominator and perpendicolarto it, and if v dénote thé operation of tuming through a
right angle round the axis p, without change of length, the
given fraction is resolved into the parts coaC+<' Nn~.K
ONTNBCMCCLPS0FQOATBMMON8.432
If the positionof thé numerator and denominatorhad beea
interchanged, it is easy to Béethat the operation of tttrnmgthrough the same angle in the oppositedirection wouldhavebeen expresaett ï cos~ sm~. Y.
6. If p, N,~8be three vectors sach that ~=a+~3, and if
F, .B represent rectangular rotationsperpendieular to thosevectors as above explained, then F'=*~+A For (see fig.,p. 858) let p=0~ <x==0~ ~9~ M, and let OP, OQ, QPbe equal and perpendicular to these lines,then if OB be a lime
perpendicular to thé. plane of the paper equal in length to 0~
It follows then that the aymbob of rectangular rotation
may bo resolved in precisely the Mmeway aa the vectomin
Art. 8; and, there~re, that if l, J, 2!' denote rotationswithout
change of length round the three axes respectivety: then asimilar rotation round an axis p, making with thèsethe angles
a, 'y, may be reaolvedinto the aamJ cos+ J cos~S+2~ coB'y.And in like manner the fraction partially reaolvedin the lastarticle may be completelyresolvedinto the sum
We see then that thé most general expression for a geo-metrical fraction ia of the form <t+~JT+c7+~S, where
a, b, c, are numerical constants. It is becaaMit can thusbe rednc~to the sumof four terme that itmcaUedt a
quaternion.
7. Multiplicationof 6'acdons, M already intimated,denotes
the sncceemveperformanceof the operationsrepreeentedby the
factors.Thos
denotesthat we firetperfbrmthe operationP V
of turning 'y mto and then that of tamiag into a, thé
Kmit being the same as if we turned 'y into a. To multiply
any two &act!oua ~tit M only necessaty to t)N'n round
in its plane until its numerator comeMewith the !nteMec6on
OSTHECAt.CUL.CSOfQUÂTERNMNfi.4S3
FF
of thé planes of thé two ~aettoos; aad until ita denominator
eoincidewith the same line, when the multiplicationh per-formed as before.
It at once appears hence that when we multiply two qa<L-ternions, the orderof the &ctorsis not mdMbrent. Thus, let
~t, -B, 0, D represent four pointson a aphereof which0 Mthe centre.TheaifweËmttHmODtoOl?
thronghan angle&,and then OJEto<?Cthrough an angle<~thé remit isthe operationof tuming OD to OC.Bat if we had commeacedwith the operationof turningthronghthe angle a, which !e that of turning OJ[ to OE, and thenOE to OB through tbe angle b, thé result îs the operationof tummg OA to OB. Now thongh thé arc JLB is equal to
CD, the plane ûfAB is generally d!Serent from that of CD,OC 0~. OB CE
MMithereforethe ptodact ~r"'?~Mnot equal to
whichis the productof two eqnal&otoMtaken in oppositeorder.If the arcs a and &be each90",then indeedthe planeof AB
willhe the same aa that of CD,but thé directionof therotationsin thé two products will be opposite. If then wemultiplyto-
gether two rectangular q~teTm&mA, B, (that is, anchthat theMta.ttonia through a right angle) we see from Art 6 thatif~.JBbe ofthe form!co<~+~mnC.~then B. Awmhe of theform coa~- mn~.P. Two quatemionsthuBrelated are aatdto be conjugatequaternions: that !a, when one !s of thé formscalar+ vector andthe other, thésame scalar the aamevector.
It followsas a part!co!arcaseof the last, that when ~=90*,the productof tworectangularquaternionswhose planes are at
right angles to eaohother, gives ~i.B'='-B. As thMis a
fundamentaltheotemwe eh&HpMMmtlyprove it Independcatly.
8. It is seen without dt<Bcnttythat the multiplicationof
quaternionsM a distributiveoperation vtz., that the prodnct
of the quatemMns~+~+~~
is the sum .f thé
OX THE CALCCLUS OF QUATENNtONtt.434
MvemtModacts &c.; and that the same th!ng !s
A At
trae if the order of multiplicationbe revemed. Hence then
if wehave two quaternions,. each expressedin the form
the product !a thé sum of the mxteen terms got by combiningeach of the firat four terms with each of thc secondfour,carehowever beingtaken to attend to the order ofthe multiplication.Let us then examine thé meaning of the terms 77, IJ, &c.,which occnr in auch a product. Now if we remember that7 denotes a reetangular rotation round the axis of x as axis,and that the effect of such a rotation would be to change a
line in the direction of the axis of y to that ofc, and one inthe direction of <! into the négative direction of y, we canwrite down the equations ~'=&,JR:'==-y. In like manner,
Jk=i, t~=–&; 7!'?===~)JEy==– Let ns now consider the
eSëct of two of these operations performed consecutivety. If
we nrst operate on with I, and then again with I on theresntt k, wegat 7y==-~ or 7*=– t. In like maBnerJ'c'–l,
JC*==--1,and since it i&evidently true, no mattcr what line
be taken for tho axis of rotation, that the effect of twice tarningrounda right angle is to reverse the positionof thé lineoperatedon; it followsthat the square of every rectangular quaternionmay be said to be -1.
Again we hâve seen that ~'=~, JT!:==t;henceJrj = i; but
~'=*–t; hence J~!=– In like manner, from the équa-tions J?=– ~:==-j, Ki=j, we conclude2y=JC Hence
Zf= jE= ~1 In like mannerj~=7== JEy ~7=. J~ Zf.If now we compare the equations -{f=~, J~='jE, &c., we
shall find that the equations which represent the encct of the
operations J, J, K on thé lines i, &, are exaclly the samein form as those which dénote the e~cts of the successive
performance of these opérations. Now since in thé practiceof this calculuswe arc concemed with the laws according to
It is aho tme, though it h not to be token for granted, that when we
take the continued produet of three quatemioM (?9*) c?(<
485ON THE CALCULU8Of QUATEBNfOSS.
FF2
whieli thé symbols combinewith each other rather than with
their interpretation, it ia found unnecemary to keep up the
distinction of notation between J, JST;<,j, k. Whatever
propositionsare tme of the symbola in thé one sense, are
cqually true in thé other, and, by interpreting some vectoraM Mnes and others as rotations, we can givo a variety of
significationsto the sameequation all of which will be equallytrue. We shall then nmierstandi to dénote at pleasure éithera unit line measured along the direction of thé axis of {e, ora rotation through a right angle round that axis. In likemanner a rectangular rotation round any unit vector a ia re-
presented by thé letter a as already stated in Art. 6. We
shall write the general form of a quaternion a+M+c/+<N!;and we shall combine these symbols according to the lawa
~=j'==&'=- ~=i!.=-Jt;~=~ M=~&.In forming the continuedprodnct of a nnmber of factors
tho order must be carefuUyattended to, except that if a scalaror nnmber ia one of thé factors ita order is indifferent, andit may be broughtto thele&hand as a multiplier of the whole.
Thus, if a, ~3,y be any three unit vectors,or rectangular qua-ternions, and if we multiply/9'yby a~ the remit e~8*'yiB ay,smce~1.
Ex.1. Toformthé squareof théunitvectori co<« eo~ t coa-y.Byactualmultiplication,~feget<*Mt~tj* COt*~+t*CM~t t ~) OM~t«MYt (Mt ?) CM'YCMe
+? +ji)cosaeoe~,
which,in virtueofthérelationsconaeeûng k,reducesto
au oaght to be the case. If the teotor be not of unit length the squareof M'+~ +&! is, in Hke mtener, (.~ +y*+ ~), or is the négative squareof the length of the line whhh thé vector represents. Ve may expressthis hy Mytag that the square of any vector ia the negative square of the<~MM'ofthat vector.
ON THE CALCULUS0F QCATERKMK8.486
îf be the angle between the two fectoM, the direction-cernes
of a petpendicular to their plane, the prodMt may be written
CMC6inC (i COSt*+j CM~" &CMV).
(Thia agreea with Art. 6.) If the vectors were respectively of tengfhsf, thh produot wouldevidently be multiplied by B'.If the product had been taken in different order the tentM part of
the product would <tiU be -eos~, but the vector part would change
Ogn. Hence, If we denote by 8 and thé operation of taking the
Ma!M and veetor part of a- quaternion, we have 'S("~) 'SQ3<t) eo*
y(</3) = ]P'(;9.). And «g«:n, we have <~ +~3'' 2N(<)).If the two vectom be at rtght angles the scalar part of the produet
eïideatty vaniahes. Hence thé condition that two vectOK e, ~9,may be
<ttttght angles M~(«jS)= 0.
Thus then if p be a variable vector passing through the origin, anda a Nxed vector, the eqMtion N (pa) = 0 may be taken as the eqaatîonof the plane through the origin perpendicular to Rince p M evidentlyUmîted to that plane.
Let it be )'eq<thed to find the equation of any other plane. Let the
petpendteuiM &om the origin on that plane be denoted in length and
direction by and let the radius vector ta any point of the plane be p,
then h the vector joining the extremity of this radim vector to
the foot of the perpendicular, and since this line is, by hypothesis, to
be perpendicctar to the equation required is ~(/) <t)a = 0 or ~(p«) = <A
But a' ia a acalar, and we may therefore divide by it nnder thé aign S,
and W)-!tethe equation ln the form= 1.
This equation may aho
be inferted &omwhat wu stated in a previous article, vis., that the scalar
part of the above fraction denotes the projectionof thé Unep on the Hnea.
In like manner the equation &~)a 1, which expresses that thé pro-
jection of the nxed line a on the direction p M in length equal to p,
obvioualy repreMnt*the tphcre described on the vector a as diameter.
Again, the equation N~) StS)cl,in the &tt place repMMnte a
cone beeause !f it M eatMned for any value of p, it will also be MtMed
for the value M!/),where m M any constant. Seennd!y, it paMesthrongh
the intmection of <S ], ~E = 1 !t is there&M the cone whose baseP
h thé arête represented by the two équationsjut written.
Ex. 3. To nnd the product of two quaternions. We have only to
muttipiy oat <t+K+~~ <?)<+M+<~ f A. We may form a dearer eon.
ception of thé result by aeparating the Malarand vector part', and writingthe two quaternions 'S'+J~ S'+~, whenthe product N &y+Nr'~ iyF+!~y*.
Now if it be required to Snd thé tcatar part of the produet (Mnce S~'`
ONTHE CUMULUS0F QUATERNMN8. 437
and ~Fare merevectort),it is SS' +N(F'F'), or the Matarof theproduetis thé produitoftheMataM+the Matarpart of theproduetofthe vectoM.
Thus let e, be three radii vectoresof a aphere; then it fa an
identicalequationthat P=*< P
Nowifa, b, c betheddMof theapherical
triangle &)tt)edby the extremitift of thèse vectoN; cMe, eM&,eoteare the MataMof the threequaternions,and the sealarpart of the productof the tecton on the right-handNdeof the equationMthe produet oftheir tensomsina, sinb,into thé cosineof the angle betweenthem, tbuswebave thefundamentalformulaofsphericaltrigonometry
eo~e=cos«cosb+sina s!n6cos C
9. We ean, in like manner, form thc product of three
vcctors. It is found, without dH&culty, by acttMtl mt!lt!pU-
cation, that if M:+~y+&<, &'+jy'+&z', <a;"+~"+~ be the
three vectors, the scalar part of the produet M the determinant
whose three rows are x, y, a; y', y, < JSatca <
a, 'y ðe three Mc~oM,the cotidition <&<ï<~ey o~oMMlie
in oneplane M ;8'(<t~y)==0(Note, p. 19).This is atso evident from the consideration that if <8'(a~y)=0
then c~8'yis a pure vector, but e~Sy= K. t8'(~S'y)+ att~'y) there-
fore <!tF(~) isapuro vector, or a is perpendicular to r(~-y),and therefore is in the plane of and 'y. Q.E.D.
Thus we can find the equation of the plane pasang throughthe extremity of three vectors a, j8, y. By hypothesia, the
Unes joining the extremity of any variable vector terminatingin the phne, with the extrenuttea of the Manmed vectom, lie
in the plane. We have, therefore, jS(p a) (~ ~8)(p 'y) = 0.
In expanding this we may omit auch terms as ~*<y,because
p* is a scalar, and p*'ya mere vector whose acalar Mnothing.The expanded product ia then
and the vectorperpendicularto the plane is F~3'y+'yw-t e~).
Retuming to the product of thé three vectors, it is ftbo
found by aettud muIttpUcatlOB,that
ONTBECALCCMJS0F QUATMNMtO.488
In connection with this, thc following identical equation
may be given,
as <~a that the vector part of the product t~8PyS may bcwritten in either of the forme
In &ct, ]~3 denotea a line perpendicularto a and j8; thovector now required must therefore lie in tho plane, both of
ceand and of <yand 8.
10. As an example of the method of applyingthis calculusto a geometrical problem, we shall investigatethe problem toRad the equation of the surface generated by a Une restingon three directing Unes. In the first place we may follow a
procès proceeding after the analogy of the co-ordinatemethods.
It is seen immediately by substituting .(&x+aM')for a in
the equationof a plane throngh three points, that the equationof the plane through the extremity of thé vectorj)Mtwritten,tmd through a fixed line, e. through the extremitiesof the
vectqrs 'y~ form L1+MB=0, where ~Ldenotea
the plane through a~-y, and B that throngh a' If thenwe join any assumed point on thé vectorad to thé other twoUneswe get the equation oftwo planes in the formM +M~B==0,L~'+m.B''=0, from which, eliminating M, we get the locusin the form ~B' =BA'.
Otherwisethus, we are to express the conditionthat, if we
join by planes any assumed point on the tocM to the three
lines, the joining planes have a line common. The vectors
perpendicular to these planes will then be co-planar. Letthen the first Une be parallel to thé lino a, and pass throughthe extremity of a vector a'; then tho vector perpendicularto the plane throngh this line being perpendicularto a andto a' –pis I~t(a* p), &ndthe required equationis
ON THE CALCCMTS0F Q~ATERXKMTS. 439
11. We give one more example to showhow In6nltesunahare introdneedinto this calculus. The equation of any apherein p'=.
Now let the Unejoining the extremity of p to an indefinitelymearpomt be <~),then the next consecutiveradius vector ia
p+< and we bave
whichindicatesthat the radius p is perpendicularto the tangentUnedp.
Very muchmore must be said if it wore mtended to giveany complète aecountof this Calculus, as, for example, themethod of 6nding the equationsof tangentsand nonnab, Unesof curvature, geoeteatc~&c. But cnoagh has been BaH to
disposethe reader to give crédit to the assertion that thereia no geometticalproblem to which it may not be applieathat !t is very rich in transformations; and that its proceMeathoagh coastandy following the analogiesof the co-ordinate
methods,ttreby nomeansBlavisbiydépendenton that system.
( 440 )
APPENDIX II.
ON SYSTEMS0F ORTHOGONALSURFACES.'
IT might be thooght, &om Dnpm'a theorem, that beinggiven a series of aar&cea, involvinga parameter,it would be
alwayspoemNeto determinetwo other systems,eachcontaininga parameter, and cutting the surfacesof the given systemat
right angles, and along their lines of cnrv&tore. This, how-
ever, is not the case. Im order that a given &milyof surfaces,with a parameter,may form one of a triple orthogonalsystem,an equation,or equations,of conditionmust be satisfied.
M. Serret arrives at the conclusion(seeZx~M?~,VoLxïr.,p. 241) that in order that the equationF (a*, y, &)=<[,wherea is a parameter, may be one of a triple orthogonalsystem,the famctMnmust Batisf~two partial differentialequationsofthe aixth order. We give Serret's investigationof the par-ticalar case where the given AmcdonM the sum of threefondions of x, y, z reapecttvety.
Let an equationthen be given of the form
It is required to determine the condition, to which these
fonctionsmust be subject, in order that the sur&ces(1) mayhave a pair of conjugate orthogonalSystems. Supposethat
are these système and it is evident by the condittons of the
problem that we have (-Y', Y', Z' being the first derived
6mctKHM of X, Y, Z)
For the following appendix, on a Bubjeetwhich I had omitted m the
preeeding treatise, 1 am almost entirely indebted to a manuecript note
Mndiy placed at my dispoaal by the Rer. W..Robettt, M well 118to hM
papNNpubIMted ln the Cb~M J~Ht/w.
ONSÏJtTBMS0F ORTHOGOKALSURFACES. 44i
Pfoeeedîng to mtegrate the 6mt two of thèse eqaattonBbythe ordinary methodsof pNttfJ d!&tent!al equations,we find
that and Y are funetionsof u and o, where
Now M and v being fonctions of x, y, we may regardy and <!a functionsof M,w, anda*. Henee x entem (8) asan indeterminateparameter,and the quantities j9 and v muet
satisfynot only (3) but alM the derivatives of (8) obtained bydMerenttatmg it on the suppositionthat x alone !s variable.
Din~rentiating (3) with respect to x on this hypothesis, and
remembering thatm rmt 'y <?t
ONSYSTEMSOPORTHOGONAL9C&PACES.442
This relation expresses the conditionthat a &mHyof sur-
faces,of the particular form representedby equation(1),shoutdform one of a triple orthogonal System. It wa~ first givenby M. Bouquet,LtoxM?~ VoLxt., p. 416, but the above proufhas been taken from M. SMMt'amemoir.
Even when the equations of conditionare s&tNËedby tm
MBnmedequation it does not Mem eaay to determinethe two
conjugatesystem. Thus M. Bouquet observedthat the con-dition just found is aa~sËedwhen the given aystemta of theform a~<c'e[, but he gave no dae to the dtacoveryof thé
eoc~tgate system. This lacuna has been completelysuppliedby M. Serret, who haa ahownmuch !ngenn!tyand analytical
power indeducing the eqttattonaof the conjugatesystems,whenthe equation of condition MBatIsSed. The actual resutts are,
however,of a rather compUcatedchamcter. We must con-
tent omaelveswith refemng the reader to bis memoir, onlymentioning the aaaplest case obtained by him, and whichthere ia no dHBcnltyin vorifyinga po8teriori. He has ahownthat the three equations,
representa triple ayatemof conjugateorthogonalSM'&cos.The
anr&cea (a) are hyperbolio ptu-aboloMs. Tho syatem (~8)is
composedof the closed portions, and the ayatem (y) of thein&ute aheets, of the mr&ces of the fourth order,
M. Serret bas observed that it followsat once from what bas
been stated above, that in a hyperbotioparaboto!d,of which
the principalparabolmare equal,the sum or differenceof the
distances of every point of the same Uneof curvaturo fromtwo Sxed generatnces is constant.
Mr. W. Roberts, expressing in dliptio co-ordinates tho
conditionthat two aaï&ceaabouldeut orthogonally,has soughtfor Systemsorthogonal to Z-~ JM+~='e, whcre L, Jh~N are
&metI<Mt9of the three eH!p~c co-ordmatesrespectively. He
ONSYSTEMSOPORTHOQOKA!,aCRFACBS. 443
haa thas added some systemsof orthogonal snt&cea to those
previonslyknown(C5~mp<MRendus,September 88, 1861). Oftheaeperhapsthe most intere«tmg,geometneally, is that whose
equation in elliptie co-ordmatesia /tty=tt\, and for whichhe has given the followingconstmctica. Let a fised pointm the Unep! one of the axes of a system of confoealetËpaoldabe madethe vertex of a series of coneadfCttmsonbedto them.The locus of the curves of contact will be a determinate
surface, and if we supposethé vertex of thé cones to move
along thé axis, we obtain a &tnuy of surfaces involving a
parameter. Two other systemsare obtained by taking pointa6ttaated on the other axes as vettices of c!tcamacnb!ngcônes.The ear&cesbelonging to theae three systems will intersect,two by two, at right angles.
It may he readily ahownthat the lines of cnrvature of theabovo mentionedsurfaces (which are of the third order) are
cu*des,whosoplanes are perpendicularto the principal planesof the etItpMida. Let ~d, J?, be two nxed points, taken re-
apect!ve!yupon two of the axes of the con&eatsystem. Tothese points twoanr&cesinteraectingat right angles wHlcorre-
spond. And thé curve of their intersection will be the locas
of pointaJtf on the comfocalellipsoida,the tangent planes atwhich pass through the UneAB. Let P be the point wherethe normal to one of the eUipsoidsat M meets the principal
plane containingthe Une~jS, and because P is the pole ofjtB in roferenceto the &)calconic in this plane, P !s a givenpoint. Hence the locus of 3~ or a line of cnrvature, ia acirolè !n a plane perpendicular to the principal plane con-
taining AB.
( 444)
APPENDIXIII.
CÎ~BSCH'SCALCULATMNOFTHES~IRFACES.'
1. Itf this appendix we give the caïcalation referred to
p. 406, by which the equation M detemuned of a surfacewhieh meets a given surface at the points of contact of Unes
which meet it in four consecutive points. It was proved,Art. 472, that in order to obtain this equation it MnecesMryto eliminate between thé equation of an arbitrary plane, and
the functioBBAU",A'!7', A'Ï7'. We performthis elimination
by solvmgfor the co-ordinatesof the two pointaof mtersectionof thé arbitrary plane, the tangent planeA!7', and the polar
quadric &*P'; aubatttutmg theee co-ordinateesacceadvely in
A*!7', and multiplying the raulta together. 1 write withM. Clebach, the four co-ordinates of the point of contact
<B,,<c,, a!j the running co-ordinates y,, y,, y,, the
dM~renëal coefficients u., «“, «,, u.; the second and third
di~ren~al -coeniclentabeing denoted in like manner by mb-
indices, aa «,“ «~. Through each of the lines of intersection
of A!7', A'~T*,we can draw a plane, eo that by suitablydeterminmg t,, te, ta we can, in an infinity of waya, form
an equation identically aatMed
We shall suppose this transformation eSected; but it is not
nocesas<yfor us to determine the actnal values of &c.,for it will be found that these quantities w!Hdisappear from
the result. Let the arbitrary plane be e~+c~-tc~+c~,then it ia evident that the co-ordinatesof thé intersectionsof
SeeNote,p.406.
Ct.EBSCH'8CAMULATMX0F THE SUBFACES. 445
tho arbitrary plane, thé tangent phne M,y,+M~+~,+M~,and A'< are the four determinants of thé two ayBtema
Theee co-ordinates have now to be substituted in A'P') which
we write in the symbolical fom (~+~,+~,+<ï~)';
where e, means &c., so that after expansion wo maydx.
sabatitute for any term ~~y~~ ".Myty~t It
ovident then that the result of substituting the co-ordmates
of the first point in A*P', may be written as thé cube of the
symbolical aeterminant S<c~ where after eubing we are
to mbstituto third differential coefficients, for the powera of the
a's as haa been just explained. In like manner we write the
result of substituting thé co-ordinates of the second point
(S~,e,M~)', (where &, is a symbot used in the same manner
as a,). The eliminant required may therefore be written
For since the quantities a, b, are after expansion replaced by
dIS~renttab, it is immaterial whether the symbol used originallywere a or b; and the left-hand atde of this equation when
expanded is merety thé double of the last expression. We
hâve now to perform the expansion, and to get rid of p and
q by means of equation We shall commence by thus
banMhtBg and q.
le The reaeon why we use a different qmpM for &o.la the M<xmd<t.Ct
determinant, is becanae if we employed the same eymbol, thé expMdedMtutt VQutd evidently contain sixth powen of a, that is to My, emth
<tif6Bï<ntMleoef&e!en<t. We awid this by the employment of different
tymbott, as m Mr. C<tytey'<"Hyperdetenaînant CatcuiM," (JE<MM<<on
jy~<r ~%f~< p. fS) with which thé method bore used M subattmtiaUytdentieat.
CMtBSCR'ttCALCPLATMN0F THESURFACE&446
The eliminant is F'+<y=-0, or (~'+6T-8FC(F+C)==0.We shall septtrately examineF+ <?,and F<?, in order to getrid of p and q. If the detennïnaats in F were M far ex-
panded as to separate the p and g which they contain, we
ahonid have
where, for example, M~in thé determinant Bot,c,M~and M~!a
S&,û,«,. If then i, j be any two satindices the coeBSdemtof
M!~ in F+ 9 M (~+j~<). And we may write
F+0'=SS~(F.~+~),
where both and y are to be given every value from 1 to 4.
But, by comparing coefficientsin equatMn we have
Now it is plain that if for every term of the formMy+~fwe ambstitute<,<~+< the resnit is the aameM if in F*and
6 we everywhere altered p and q into t and u. But if inthe detennmtmta S<t,c,«~, B~,c,M~ we alter into «, thedetermmaats would vaniah as having two coIamMthe same.The latter set of terms thereforein F+ <?d!aappeaM,and wehave i (F+ 6~ =SS~~M~.
Now if we remember what M meant by m~ thm doublesnm may be written in the form of a determinant
For smcetins determinantmuât contain a constituentfrom each
Ct.EBSCH'S CAMUt~TtON OF THE SURFACE 447
of thé last three rowsand co!umMit is of the &mt degree m
M, &c., and the coeS~tentof <myterm «~ is
In the determinantjust written the matrix cf the HesNanis borderedverticallywith<<«; and horizontallywith h, c, u.As we shall have frequently oecaaïonto ~tseAetennmmtBofthis kind we shall find it convenient to denote them by an
abbrevtatMn, and shall write thé result that we have juetarrived at,
3. Thé quantity F<? is transformed in like manner. It
ia evidently the produet of
Now if the first Une be multiplied out, and for every term
(~A+MJ we substitute its value derived fmm equationit appears, as be&re, that the terms including t vanish, and it
becomes ~Bmr~u~, which, as before, is equivalmt toC4 C~
becomea SSM)~ which, as befbro, ia équivalent to t~ ),1
where the notat!on indicates the determinant formed by border-
ing thé matrix of thé Hessian both verdcally and honzontaHywith a, c, «. The second line is transformed m like manner
and we tbus find that (F+ 0)'-8~(-F+ <?)<=0tmasfbrme
into
It remains to complètethe expansion of this symbolicalex-
pression; and to throw it into Mch a form that we may beable to divide out e,iB,+c~)!,+c~+< We ahaRfor short-ness write a, &,< insteadof a,.c,+<t~+o~+<~a;~ ~a;,+&c.,<f,+&c.
CLMSCH'SC.U.CtJLAT!<Kt0F THE SURFACES.448«tthiadeterminantmay be reduceAby multiplying thé Sjst fow
columns by a* a* a' a' and subtracting their sum from the
!ast columnmultiplied by ("-t)) and stmUarty for the rowa;when it becomea
where(~]
dénotes thé matrix of the HeMian bordered with\o~
a single line,vert!cal!yof s'a andhorizontallyofb's.In like mannerwehave
Now as it will be our Crat objectto get rid of the letter a,we may make thèse expressionsa little more compact by
writing ~-6e,=~,&c., whcn it is easy to ace that
CLEBSCR'SCÀÏ.CULATïON0F THE SURFACE <& 449
a<ï
8. We proceednowto expand and anbatttute for eachterm
<t,o,<t,,&c. the correspondingd!ËhroNt!fJcoefficient. Then, inthe firstplace,it is evident that
Bat tho last determinant is reduced as in many similar
cases, by subtracting the first four columns multiplied re-
spectively by y,, a; a; from the fifth column, and M causingit to vaniah except the last row. Thus we have
Lastly it is neceeMo'yto calculée a( ) (j)
Now if P~
dénote the minor obtamed from the matnx of the HeMt&n by
eraaing the line and column wh!ch contama M~; it le easy to see
that<V~==-(tt-2)S~P~M~c~,
whera the nambem
M, M, q are eaeh to recetvo in tnm all thé values 1,2,3,4.
Bat, aeeZ~soM~on Bt~e~ ~~e&M, Art. 28,
CLEB8CH'8CAÏ.CUMTKWOPTH<!M!RFACE&4M
Bat attending to the meaning of the eymbok <<“&c., we see
that d or <~<B,+<B,+<~a',+o~ vanishes identically. If thenwe ambe~tutein the equation wMchwe are reduoingthe values
jast obtained it becomead!viaH)kby e*, <mdis then broughtto the form
But (Art. S) the hmt term in both theee can be reduced to
t2 (M 2)'JT*c Sabtractiag then, the &ct0f avides oat
agtdn, and we have thé finalfMalt clearedof irrelevantStetors~expressedin the eymboUcal&)ïm
7. It remains to shew how to express this result in thé
ordinary notation. In the CMt place we may transformit bythe identity (seeArt. 76, andZeMOMonJ~~er ~~o, Art. 28)
CLEBSCH'8CALCULATIONOF THE SURFACE & 4M
aaa
Now( ) (j (t.) expresses the covariant which we have befoM
called 0. For giving to C~,the sMnemeaning ae before, thé Bym-bolical expression expanded, may be written B~~O~M~wttere each of the suiBxeBis to receive every value from 1
to 4. But the differential coefficient of ~Twith respect to
can easily be seen to be SC.K~ M that 0 is S~ t~r W
Il
wh!ch ie, in another notation what we have called 0, p. 899.
The covanant jS is then reduced to the form 0 4B<&,where
where 0~ dénotéea second minor formed by enuaag tworowsand two coinmns&om the B)atr!xof the Hesman,a form
scarcely so convenientfor ealculat!on as that in which 1 had
written tho equation,~%tZM<~A<ca~2~ttM<M<M<M,1860,p. 389.
For surfacesof the third degree Œehschbas observed thatreduces,as was mentionedbefore, to S!~H~, where de-notesa seconddifferentialcoe~Ctentof B.
8. If at any point on a surface &o<&inflexionaltangentsmeet in four coincident points, either of the intersections of
AP', A'!7' and an arbitrary plane must satisfy A'P'. Forsnch points then the expression at the end of Art. 1 must
vanisb, as Mon a9 we have made the ~ymboucatsubstitutionfor a, independentlyof any suppositionas to the vahMof theb symbols. In the equation then which we found at this
stage of our work
we may consider both the b and c aymbole which oceur in d
aa arbitrary constants, and the equation just written whose
degree in the variables is easily eeen to be 10a-Ï8, ahowathat ~ot<y& thepointe OKa MO~ce where two domblyM~ea!M)M~
tangents can drawn, or, in other words, through the pointswhere tS*and <7 touch, OMK~at'<~ of Mo~tCMcan p<Mtof <~
degree tOK -18. (See Art. 476.)
Ct.EBSCH'8CÀKtHjLMONOPTBE8UBFACEA4&2
To find the points on a anr6tce where a Une can be drawnto meet in five consccuttve points, we have to fonn the
condition that the intersection of A!7't A'P\ and an arbitraryplane ehould eatta~ A'U', as well as A'C". M. Ctebsch has
applied to A*!7' the aame symbolical method ef eliminationwMch bas been hem appHed to A'P'. He haa Mcceeded in
dividing out the factor e*from this resuh: but m the finalformwhich he bas &nmd,and for which1 refer to Ma memoir,there romain c symbob in the aecond degree, and the resalt
being of the degree 14M-3& in the variables, all that can bec<aidodedfrom it is that through the points which 1 havecalled a, (p. 408) an infinity of satfaces tan be drawn of the
degree 14M-80. We can say therefore that the number ofmch points does not exceed M(1ÏM-24) (14n 80).
9. Me <M<~teeS <otMJ5Mthe <tM~aceH along a eey&t~eMt~e. Smee the equatton j8 is of the form @-~LB<&=0,it in anntCtetttto prove that 0 touches But since @ is gothy bordermg the matrix of thé Hessian with the d!<Rsreat!aÏ8of the HeMian~0~0 is equivalent to the symbolicalexpmsNon
)–J=0.But, by au identical equation already made ase~of,
wehave
,where c is arbitrary: Hence 0 touches H along its inter-
section with the snr<aceof the degree Ta–15,( /KT\).
It M<~
proved then that N touchesB, and that throngh the eurve ofcontact an infinity of sar&ces can paes of the degree 7n 15.We bave made use, p. 418, of the theorem that the curvesPS and ÏTB~toneh each other.
( 4M )j
APPENDIX IV.
ON THB OBDEB.0F SYSTEMSOP EQUATIONS.*
1. WE have showed, p. 250, how to determine the char
racteristics of a curve given M the intersection of two surfacesbut it bas been remarked (p. 28&) that there are many curves
which cannot be so represented. There M no algebraic curve,
however,which may not be represented by meau of the equationsof a system of surfaces beeause (p. 240) by taking m large
enongh we can always find a namber of surfaces of the <?'"degreeeach of which shall entirely contain the curve. But any two
surfaces of thé system will not &/?M thé curve, for their
întersection will in general consist of the curve in questionand an extraneous curve besides so that the curve is usuallynot the complete intersection of any two, but only that partof the intersection which is common to ail thé rest. Thé
object of this appendix ia to show how, when a system of
equations is given denoting surfaces which pass tbrough a
common curve, thé charactenstica of that curve can be de-
termined.
In like manner if we are given r points in space, we eau
atways, by taking m large enougit, determine a number of
surfaces of the M"' degree whieh shall pass through the given
points. But ordinarily the intersection of thrce snch sor~ces
will consist of the given points and extraneous points besides;and we cannot <&~e the given points except by a system of
more than threo equations, the given points being the ontyones which satisfy all the equations. Conversely, it is tbe
object of this appendix, when snch a system of equations is
given, to ascertain the number of points which satisfy a!
Bee Qt«tf<<f~~e«nM<o/J&~tma<)<'<, Vol, p. 246.
ON THE QBDER 0F 8TBTEM80F EQUATtONS.454
2. The eimpteatillustration of this is to take four planes&+\<t)&+X~c+~<!+\6; wheîeo,a~&c.rapreaentpltmes,and X is an indeterminatocoe&c!ent; then if we form the
condition that theee four planes ahouldmeet in a point, this
eomdi~onia known to be of thé fourth degree in X. It Mtow*that four values of can be found for which these eqaa~omwill t~reBent planes meeting in a point. And obvioualythefour points 80 found most aatiafy any of the aix eqoatMNM(snchaa <=~a), which are got by eliminatingX betweenanypair of the given equations. Yet these all represent surfacesof the second degree, any three of which intersect in e~A<points. It followsthen that the ayatemof equations
denotes a system of surfaces having four pointa m common-1but that any three sur&ces of the system intersect not onlyin these four pointsbut m four extraneouspoints. In generalthen, supposewe are given ~+8 equationsinvolving f para-meters, it is evident that by eliminationof the variables we
get a sunictent nnmber of equationsto détermine systemsofvalues of the pMtuneters for which the equations will denote
surfaces having a point in common. It is evident aleo thatsach points must s&tisf~the equationsgot by eliminating the
parametera between any ~+11 of the given equations. Andyet any three of thèse latter equations will dénote surfaces
iaterseoting not only in these pointe commonto all but incertain extraneous points besides.
9. In like manner if we had been given the three planese+X<t, ~+\ c+~y, it is obviouBthat we may give to Xan inËnity of vaines, to every one of which oorrespondsa
point whichis the intersectionof the three correspondingplanes.It is obvious abo that thé locas of all those points must bea ourve commonto all the surfaces<~8-&a,~-e~, cet-o~.But it waa proved, p. 241, that though any two of thèsesurfacesintersect in a curve of the fourth degree, there is
only a cuMceommonto aU three. And in general if we are
ON THE OBDM 0F MtMMS OF t~UATMNS. 4M
given f+3 2equations,involving r parametem, an infinity of
eyatemsof values of theee patametetwcan be determinedfor
which tho equadomawill dénote surfaces having a point in
common. The locus of theae points will be a curve, which
will be commonto aU the sarfaceagot by eliminating thé
pammetersbetweenany f+11 of the equations. Yet any two
aoch surfaceswill intersect not only in this carve but in an
extraneous carve. Let os suppose then that we have f+11
equations, involvingr parametem in thé first degree. Thé
eUmmattonof thèse gives rise to a system of determinants
where the numberof horizontalrows is aupposed to be y, andvertical y+1. We propose to determinethe charttctene<icsof
the carvo whichis commonto the mtr&cesfepresented by aU
theae determinanta.
4. To Sx the ideM we take the matrix with four row9
and five columna
the methodof proofnsedin thia casebeing generally applieable.We suppose the funetionso, t, &c. to be of any degree, but
we supposethe degreesof the coKespondingfonctionsin either
the same row or the same coinmnto be equi-dt~erent. Thus,if the letters <t,b, c, &o. aiso indicate the degrees of thèse
fonctions,we supposethe degrees of d, b', &c. to be a+a,
6+a, &c., of d', b", &c. to be a+~, &+j8, &c., &c. Let
P and Q dénote thé snm of the qnantities a, b, &c., and the
snm of their productsin pairs; and let p and q denote the
correspondingsnma for the quantities a, /3: then we assert
that the order~<~ CMMMcommon<0<tB <M)~!tCM<'<pfMM<e~<Xe<&<eMMMMt!&0/'<~ <y<<MMia ~+jpP+~g<In the Sr6t place we observe that if this formula is true
0!t THEOM&BBOPSYSTEMSOPEQCATîn~456
it Mlowa that if the ordem of the tunctione&,V, & &c. hadbeen a+a', «'+«', a"+< those of c,< o+~ <t'+~S*,&c.;and if we denoted the snm of the qaant!t!eB o', a", o" andof their products lu pairs by P', and the eorre<pondmgMms for a', ~3',<&c.by p', g', then the order of the curre in-
vesttgated would be +j)'F' + F~ ~'< For the fermaof the
top row being of the degrees <t,o+a', <t+~ a+'y', o+o'; ithe q<Mmtitieawh!oh we.have caUedP and Q are reepeetivety6a+j?', and 10tt'+4op'+gf'; wMIe, a!nce the dMerenceebe-tween the orders of the Srst and succeedmg rowt are
a'–<t, <t"-<t) a"<t) it 6)Uowathat the quantMes c&tledpand g are f 4~, 3i~a + 6a', and on substituting thèsevalues in ~+~F+~)* we get g'+~.P'+.P" And in
general if tho nnmber of rowx be k we have
6. To estahUsh now the truth of the formula it is sn&Ctent
to show that it îa troe for a system with rows if it is true for
a system with &-1. It is easy to eeo that the curve we are
considering is part of the intersection of the sur&ccs denoted
by the two determinants (efc"<r"),(<t&'o"e'") and that thèse
two surfaces both pass through the carve
which does not lie on thé surfacea represented by the otherdetennuMmteof the system. Now if the aam and aum of
productsin pairs of the quantiticsa, b, c be denotcdby P",it ia easy to see that the order of the two determinants is
F"+p-t<~ JP"+F+e. WhUe the order of the irrelevantcurve is, hy the last article, g+~P"+J°" Subtractingthis number then from the product of thé other two, weget
OKTHE OB&ES0F SYSTEMSCF EQUATIONS. 4S7
or to J~-t~. Now thé truth of the theorem is eaeUyeeen for a matrix of two rows and three columns, therefore
it is generally truc. It Is needless to remark that the formula
wa9 at first obtained by commencing with the simpler caBO
and proceeding on to the gcnerat one.
When aIl the rows are of the eame order; we have a, &c.
all = 0, and therefore p, 9 both = 0, and the order of the
system is Q.
6. Next let it be required to Sud the order of the develop-able generated by the curve cousidered in the preceding articles.Let B be the sum of the products in thrces of the quantities
<t, b, c, &c. (Art. 4), and the con'espondtng sum of thé
qaaatttte: a, ~3, 'y, &c., then we say that the order of the
developable in question is
In the 6mt place, admitttng the truth of thm formula it fbUowa
that if JP', had been usod with refrénée to a, < <t",a'"
the orders of the terme in the first vertical row, &c., then the
capital and amaUletters in the formula would simply be inter-
changed, and thé order of the develop~Me would be
This ia provedexactiyas in Art. 4.
Again, we know, p. 253, that the ranks of two systemswhich together make up thé !ntcHect!oaof two sur&ceBarecoNnectodby the relation
And if these substitutionsbc made and thé rcsult reducedbytheidennties
ON THE ORMN 0F SYSTEMSor EQUATIONS.468
It followsthen that if the fbrmnh be trae for a matrix with
k rows,it is true for one withk+1 and mmceit is eMHyprovedto be tme for three rows, it b generallytrae.
7. Let us next coamdera matrixeach as
where the Bomber of columns exceeds the nnmber of rows
by two, and let M examine how many pointa are commoa
to aIl thé smr&cesrepreBentedby the determinants of the
system. Now any thMe surfaces (a&'e"<f"),(o6W"), (<<;y")bave commonthe carve
and if m, n, p be the degKes of the sar&ceB, Md p tho
degree and rank of the curve, then (see p. 2&8)the surfaceswill intereectin pointsnot on thisearve, in number
The capital and émaillettere would be mteKhanged if we
had aaed Il in referenceto thé lettem in the Stst column
<«', e",<t' If the aeventitwa had been of thesame
degMes,that iB,if a, &c.t!l '='0, then the number of pointsrepresentedby the syetemts
ONTBBOMBR0F 8T8TEM8Of EQUATtONS. 459
8. It may be deduced hence that the sor&oe representedby any symmetncai determinant bas a detemmmte number ofdoublepoints. Let the snm, sum of produet8 in pairs, aad
aum of products in threes of the degrees «f the leading terms
a, e~, o~, &c. be P, Q, JK,then the nnmber of snch double
pointe ia ~(F~).Now we have the identical equation (ZeMMMon J?~
~~Mt, Art. 28) ~(~)'==(7D, where means the
minor obtained by erasing from the given determinant the
Une aad column containing a, D is the determinant itself,and C is the second minor obtained by erMing the two Unesand colamM which contain o, <t~. Now it is évident thatthé sor&cerepresentedby ~(~)' bas as double pointsthe intersections of J~, jl, and the degrees of these
being respecdvely J'–a, P- b, JP–~(<t+&), the number ofdoublepoints is the product of thèse three numbers. Let the
anm, and aum of product of pairs, of the terms exdasive of <t
amd be denotedbyj~ g", then the product
These are then double points on the complex system CD;and are therefore either double pointa on C, double points on
D, or pointa of intersection of C and D. Now if we eramfrom the matrix thé firat two rows, aH the determinants ofthe remaining system (of which <7is one) have commonanamher of points, which can be calculated by the formula ofthe last article, by writing ~(<:+<!),~(c-t-&), &c., ~(J+o),
(d+b), &c.,for the degrees of the rows. Thé reenitM
But the pointswhosenumber bas been just &)andare pointsat which touch, and they each count for four
Maomgthe inteNectMMof thèse enr~ces. SabtrMtlag thenfour times the number jmt found from the total number of
mteraections,we get
ONTBE ORNBBUFSMTNMSOf EQUATMN&480
wheace we team that if the number of doublepoints on theaudace repreaented by the eymmetricat determinant <7 i.
~O/V-f"), that of those on the surface D ia ~(f~),and the first theorem bemg established in the simplest case
the other Mgenerally true.
9. There is stili another questionwhich m&ybe proposedconcerning the curves, Art. 4. Let there be four surfaceswhose degrees are X,, X~,and whose coefficientscontain
any new variable in the degrees tbenthe eliminantof these four equationscontainsthe newvariable in the degrce
Now X, are thé orders of the curve of intersection
of the first and second, and third and fourth sar&ces re-
specdvely; and if we eaU ~+/ /+/ the weightsof the same curves, we can assert that the weight of the con-
dition that two curves may intersect is the snm of t!te productaof the weight of each curve by thé order of the other. Now
we hâve seen what is the order of the carve denoted by a
System of determinants, snch as Art. 4 it remains to enquirewhat ia the weight of the same system. It is easy to sce
that when a curve breaks up into two simpter curves the
weight of the complex curve !s equal to the sum of the weightsof its components. We may therefore proceed as in Art. 4,and the following is the result. Let the fonctions a, b, c, &c.
contain the new variable in the degrees A, C~&c.; a', & &c.
m the degrees ~+c[', J?+a', &c. Let .P', Q', denote the
smn, snm of products in pairs, &c. of the quantities A, B, &c.
and letp', q', &c. denote the corresponding stms for a', ~8*,&c.:
let t8 denote the sum 2 (aB) where each a is multiplied byall the capital letters except A, m that j8'iaa!.M'~jRP'-S(<t~).Let also < = S(a~8'),which is in like manner~p' S («a'). Then
the weight of the system is
OKTHEORMHtOPSYSTEMS0FEQCATÏ&N8.461
If we had used P to dénote the aum of the degreee in theSrat column tnstead of in the Srat row, &c., tben thé capitaland «maU lettem in thé preceding formula would bo inter-
changed.
10. We proposenext to investigate the order and weightof thesystem of conditionsthat the two equations
may bave two commonroots. It is evident that ïn orderthatthis ehooictbe the case, two conditionsmust be fntSHed;andif t be a parameter, and a, &c. functions of the co-ordmates,thèse conditionswill represent a curve in apace. But in pointof &ct, we obtainnot two, bnt a syatem of conditions,no twoof which BaNceto df/&)ethe given curve. Thèse conditionsare (ZeMMMon .B~ef AlgebratArt. 88) the determinantsofthe system
where the &t)t Ime is repeated M–l 1 timea,and the second
m–1 1 times there are <)t+K–2rows, and M+a–1 columns.
The problem is then a particnlar case of that considered, Art. 4.
We suppose thé degrees of the fumerons introduced to be
eqn!-difEBrent that Mto say, if the degrees of a, a' be we
suppose those of 6, b' to be X+a, ~+<t; of c, c' to be \+2a,
/t+8a, &c. To find the order of the system, we nue the
formula (Art. 4) g+~JP+F* M the sum of M+tt-8 8
terms of the acnea 2a, 8<x,&o~ and is tbeM6)M, if ~re ~mte
M+w=~, 1n the same case q is the mm of
prodacte in pa!m of thèse qnamt!Uee,and Mthefe~M
ONTBtBOMEtt0F SYSTEMSOf BQPATMNtt.462
AgamPistheeam ofa–1 tennaofthe eeneB~ \-ft, \–9a,
&c., an4 of <K-ï tenns of the series ~-cc, ~-2<t, &c.
We have then
If the eliminantofthe equations
represent a snr&ce,the c!)rvehère consideredis a double curveoa.that surface.
If all the fnmctîonB<ï,b, &c. are of the &'at degree, thésurface generated is a raled surface; and wntmgX='='l iand a = 0 in thé preceding &nnata, we fmd that the orderof thé double curveis (<a-m-1) (<?+K 2).
If the two equations consideredare of thé same degree,that is to say, if <?<=?,we may write X+~=~ ~=~ and
the same formulagivesfor the degreeof the doublecm've
11. We can in !!ke manner determine the ordor of the
system of conditions that the eqoat!ona <+&c., o'<*+&c.
may hâve three commonroots. When geometrically inter-
preted thèse conditions represent triple points on the sar&ce
represented by the eliminant of the two equations. The con-ditions are representedby a systemof determinants, the matrixfor which Mformed as in the last article, save that the line
ONTHEORDEBOP&YNBM8<? BQUATtONN. 463
a, b, c is repeated M-*9 limes, tmd the Unea', c', Nt-a2
times; and the matrix consiste of M+M-2 eotnmM andm+«'-4 Mwe. The order of the system N cdcalated &~mArt. 7, and Mround to be
The order of the developablegeneratedby the doublecorve
(Art. 10) is calcutatedin like manner by the formulaof Art. 8,but ihe number so found must be reducedby four times the
nnmberof triple pointsjust found,whichare alM triple pointson that curve. Thus in the case of the rnled eoî&cethe rankof thedoublecarve is 3(M+M–2) (m+M-S).
To find the weight of the same system we have only to
applythe mme method to the formulaof Art 8. Let the terma contain the variable to be eliminated in the degreeX. andthé anelimmated in the degree X',and let the terrns A,o, &c.
decreMeregularly in the former and Increaaem the latter;ao that their degrees are ~-1, \-8, X'-l, ~3, &o., then
the weight of the system !a
12. The next system we disccss M that formed ty the
ayatemcf eondi~OMthat the three equations
<~+~'+&c.=0, «'<<T''+&e.~O, <tV+y'<+&c.=~
may bave a commonfactor. The system may be expressed
ON THE ORMS 0F SYSTEMS0F EQnATMKH,464
by the three equations obtained by clumnating ia tam be-tween every pair of theae equations, a System equivalent to
two conditions. Systems of equations of lower degree can be
got by multiplying the given equations by t, < &c. until there
is a auSeient number to eliminate dialyticaUy all the powersof t. The order of the system may be fbund by eliminating from
the equations x, y, wh!ch enter implicitly into «, b, c, when
the order of thé resulting equation in t determines the order
of the system. Let us suppose that the orders of a, <t',<t",are
X, ~t, f respectively and of b, b"; ~1, ~-1, &c., then1 found (~tMf/e~ Journal) that the order of the system is
the value for the weight however haviag been only obtained
by mducëon.
13. It is a particular case of the preceding to find the
order and weight of the system of conditions that an equation<t<'+&<+&c. may have three equal roots; hecause these
conditions *'re found by expressing that the three second dinc-
rential equations may have a common factor. Writing in the
preceding for ?, M, and M,M-2; for~t, \-1; and for f, X-2,we and for the order of the syatem
8(M-2)\(~-)t)+M(tt-t)(M-2),
and m like manner for its weight
6(M-2)\X'+8ttfM-2)(\-X')-2K()t-l)(M-8).
Again, to find thé order and weight of thé systemof cën-
ditions that the same equationmay have two distinct paiMof
equal roota we formfirst, by Art. 10, the order and weightof the system of con~itiomBthat the two first diSerentMda
o~'+&c., ~f'+&c. may have two common factors. We
subtraci then tho orderand weight of the systemfound in the
firat part of this article. The resuitis that the orderis
465ONTM ORDEB0F NTSTEMS0F EQUATIONS.
The &rmuka of thia article are those of which use hae beenmade p. 407. It would be deforaMeto Sud in Ukemanner
the otder sad weight of the system of conditions~h&tthree
curveashoaldhave two pointecommon, that four carveashoatd
meet in a point, that a enrve ahotddhave a cnap, or t~o pairsof doublepoints, &c., bnt theae problem have not yet been
solved.
THB END.
w. tCttMim, Mnrctt, Otoot oEMOBr,ttXMiMtt.
BYTHE8AME AUTHOB.
aA TREATISE ON CONIC SECTIONS.
nw xa;o~. Prieelu
LmmoN t LMKHttN Atm Co.
A TREATISEONTHEHMHEBPLANECURVES.
P)cîcel2t.
DCBMN!HOD9MANBStC~B.
LESSONS ON HIGHER ALGEBRA.
Mee 6<.
DcBLCt: HoMM ANB8'nra.
SERMONS
PBEACBEDIN TEECBABEL0F TRBMTÏCOLLEGE,DUBLIN.Mce6<.
LmmoNANDC*jmuMt: Mt<wtT.t.*nA)mCo.
