Ganguli, Theory of Plane Curves, Chapter 10

8/14/2019 Ganguli, Theory of Plane Curves, Chapter 10

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CHAPTER XRATIONAL TRANSFORMATIONS

198. In Chapter I, we have discussed particular methodsof transformations, such as Reciprocation, Inversion, Projec-tion, etc. and thereby from known properties of one curvededuced those of another derived from it by any of theseprocesses. In the present chapter, however, we shall studythe general principles of these methods, which consist chieflyin instituting a relation between any two points P and P'in the same plane, or in different planes lying in a commonspace. The case of different planes properly belongs tospace-geometry, and consequently, without any reference tospace, we shall regard the planes as superimposed one uponthe other, so as forming a single plane. Thus we shall haveto consider two figures consisting of points, lines, etc., in thesame plane, instead of two figures in different planes.199. RATIONAL AND BIRATIONAL TRANSFORMATIONS:If ( o r , y, z) and ( : 1 : ' , y', e') be the homogeneous co-ordinates

of two points P and P' in two planes (or same plane) a and {3respecbively, then the transformation, expressed by theequations

x' : y' : z '=f,(x, y, z ) : L!, '1 / , z ) : fs(x, y, z ) (1)where fll f., fs are known functions of x, y, z, each of ordern (say) without a common factor, indicates that to anysystem of values of : 1 : , y , z, there corresponds a single systemof values of a),y ' ,e' ; but to a given system of values of x ' , y', z 'there will not, in general, correspond a single system but afinite number (D~l) of values of x, y , e.


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250 ~HEORY OF PLANE CURVESH, however, D=I, i.e., if to a given system of values of

w ' , y ', z' there corresponds a single system of values of x, y, z,expressed by-w : y : z=F1w ' , y', e') : F,(ai', y', z') : Fs("'" y', z') ... (2)

. F i,Ft, Fs must also be rational and of order n, and whensuch mutual expression is possible, the relation is called aBirational TransfCYl'mation,.or Cremona Transformation, sinceit was first studied by Cremona. ' * '

200. LINEARTRANSFORMATIONS:DEFINITION: Any transformation by which two figuresare so related that any point and line of one correspond to one,

and only one point and line respectively of the other, andconversely, is called a linear homographic transformation.Since the correspondence must be (1, 1), the required

expressions cannot contain any radicals.Thus,where i..fl' fs are algebraic functions and polynomials inw , y, l I S , having no common factor.Since, to any straight line Zx'+my' + nz ' =0 corresponds

which must be a right line for all values of l, m, n,the functions fllf.,fs must be linear in x, y, z ,

whence x, y, z can be expressed linearly in terms of ;v ', y', z ',Hence, assuming proper triangles of reference and the

ratios of the implicit constants, we may write, without lossof generality, w ' : y' : z'=:11: y : z . Cramona-Bologna Mam (2) Vol. 2, (1863) p. 621 and Vol. 5

(1864), p. 3, or, Giorn. di mat. Vol. 1 (1863), p. 305,Vol. 3 (1865),pp. 263 and 363.


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RATIONAL TRANSlIOR1.UTIONS 2inThe above transformation contains eight independent

constants, and consequently, any four points (or lines) of onefigure can be made to correspond to four points (or lines)in the other. Therefore this transformation leaves thecross- ratio of any four elements unaltered, so also the orderor class of a curve remains unchanged.

The method of projection explained in . 12 is aparticular case of linear homographic transformation, whichinvolves only five constants, the vertex and axis of projectionreducing the constants by three.

201. COLLINEATION:The linear transformation admits of a double interpreta-

tion. It may be regarded as a transformation of co-ordinates,or as a relation between the points of two different (orsuperimposed) planes. Let us imagine that the planes areinfinitely near one another (or superimposed) and supposethat the points are referred to the same triangle of reference.If then P(x, y, z ) and P'(u', y', z ') represent points

in the two planes respectively, the linear equations-x'=ax+by+cz " IIy'=a'x+b'y+c'z J 1)z'=a"x+b"y+d'z

establish between the points of the two planes an (1, 1)correspondence, which is called " linear affinity" orcollineation,'*' such that to each point P of one plane corres-ponds a point P' (point-image of P) in the other. Thisrelation, however, is not reciprocal, i.e., to the point P'does not, in general, correspond the same point P, butthe corresponding point in the first plane is obtained by

This was first discussedby Mobius-Barycentric calculus (1827)


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2 5 2 THEORY OF PLANE CURVESsolving the equations (1), provided the determinant D,. ofthe co-efficients does not vanish.Thus, D,.~=Av' + By' + Cs'

D,.y=A'x'+B'y'+ C'z'D,.z=A"J;'+B"y' +C"z'

where A, A', A", etc. denote the minors of a, a', a", etc.'I'hese formulee are evidently the same as for the transfor-mation of co-ordinates, where the variables are theco-ordinates of the same point referred to two triangles ofreference, while in the present case, they are the co-ordinatesof two different points referred to the same triangle.Thus it will be seen that if P describes a curve in one

plane, P' describes the corresponding curve of the sameorder in the other plane, and in particular, a straight linecorresponds to a straight line, a range of points or pencilof lines corresponds to a projective range of points or pencilof lines respectively.

202. COLLINEATION TREATED GEOMETRICALLY:The geometric determination of collineation is contained

in the following theorem :If to four points of one plane, no three of which are

collinear, there correspond in the other four points, no threeof which are collinear, the linear relation, i.e., collineationbetween the points of the two planes is completelydetermin ed.For, if we are given four pairs of corresponding points,

the equations (1) of the preceding article are uniquelydetermined.The following geometrical construction is useful and

interesting:To the line [oining any two points of one planeIIcorresponds, in each case, the line joining the two

corresponding points of the second plane l., and to the


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RATIONAL TRANSFORMATIONS 253point of intersection of two lines in ~ 1 corresponds the pointof intersection of the corresponding lines of ~.. Now, thefour points of ~, determine six lines, which again determinethree new points, namely, the diagonal points of thecomplete quadrilateral. To these then correspond the threepoints of the plane ~. obtained by a similar construction.If now the lines joining the three points of ~, are producedto meet the six sides of the complete quadrilateral, weobtain new points of ~, whose correspondents in ~. areconstructed exactly in a similar manner. Thus, the entireplane will be covered over by a net-work of lines, and bythe continuous crossing of the meshes of these nets, we shallobtain a point indefinitely near to a point of ~" If thecorresponding nets are constructed in ~t' the respectivecollineation of the individual points of the two planes isestablished by the nets, and consequently the collineation iscompletely determined."

20 .3 . THE DUALISTIC TRANSFORMATION: tWe have thus ar considered only linear transformations

in which a point corresponds to a point and a line to a line;but there are transormations where a point corresponds to acurve, or example, in Reciprocation a point corresponds to a.line and vice versa. Such a transformation is called "SkewReciprocation" or " Linear Dualistic Transormation."Reciprocation as described before is a special case othismore general linear dualistic transormation and differs fromit only by a linear transormation.Let (x, y , z ) be a system of point co-ordinates and a , 7], C )II. system of line co-ordinates in the same or in different planes.

Then, a point in the first system corresponds to a.line in thesecond, if the co-ordinates of the point are proportional to

< Mobius-Bar. Cal. p. 273. For analytical treatment, the studentis referred to Scott-Modern AnalytioalGeometry 223226.t For a detailed account of the theory, Bee Salmon'! H. P. ourves


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254 THEORY OF PLAu~ CURVESthe co-ordinates of the line, i.e., x: y : z=t : YJ: t andconsequently. to any line lx+my+nz=O corresponds thepoint l~+mYJ+nt=O.In the general dualistic transformation, however, the

co-ordinates of a line are functions of the co-ordinates of thecorresponding point, and the transformation is linear whenthole functions are linear.Thus, ~=a\~+bly+CIZ " ')IYJ=a.x+b.y+c.z I (1)t=asx+b.y+csz )

where to a point (x, y, z ) there corresponds the line U , T f . ~ )in the same or different planes. 1,however, we put-x': y' : z'=al;c+b1y+C1Z : a,x+b.y+c.z : asx+bsY+ca!:

i.e., if a point (x', y ', z ') is obtained corresponding to thepoint (:1 ', Y , z ) by a linear transformation, there is a.correspondence between the point ( :v ', y', z') and the line( e , YJ,t), and we have, as stated above, the followingrelations: x': y': z'=~: T f : CThis shows that the systems (x', y', e') and (~, T f , C ) are

reciprocal with respect to the auxiliary conic ;c.+y' +z =0.Thus, the linear dualistic transformation differB fromthe interchange of point and line co-ordinates only by acollineation.204. 1 we solve the ' equations (1) of the preceding

article for x, 3/,z , we obtain the following relations:~x=Al~+A.Tf+Ast IJ (2)

where AI' El ete., are the minors of al bll etc., III thedeterminant 1 : 1 of the co-efficientsin (1).


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RATIONAL TRANSFORMATIONS 2 5 5The system (2) is said to be "dual" of the system (1)Now consider the point (m', y ', z') in the first system.

Its corresponding line in the second system is then :-:c'(alx+b1y+C1Z) +y'(a.x+blly+VllZ)

+z'(asx+bsY+c sz)=Oor m(al: ! : ' +a,y' +nsz') +y(b1x'+b,y' +bsz')

+z(c1:r'+CSY'+vsz')=0 (3)The equation (3) expresses the relation between any

point (x', y', z') of the first system and any point (x, y, z) 0:1a.corresponding line of the dual system. If now (x, y, z ) isconsidered fixed and (x' y' z ') variable, we have for th",line of the first system, corresponding to any point of thesecond,

x(a1x' +bly' +v1z') +y(allx' +b.y' + c1z')+z(asx'+bsY'+csz')=O (4)

The lines (3) aud (4) do not, in general, coincide j hence,in the general dualistic transformation, every point has adifferent corresponding line, according as the point isregarded as belonging to the first or to the second system.

The conditions that the lines (3) and (4) should coincidegive three values of (x', y', z'). Hence there are three pointsin the plane associated with their corresponding lines in adefinite way, regardless of the system to which they belong.

One of these points, however, is given by-x' : y' : z'=c. -bs : as -V: b1 -all

and the other two are real or imaginarj .The two lines (3) and (4) will coincide for all points of

the plane, if for all values of x', y', z ', we haveal:c'+b1y'+C1Z' : allx'+b.y'+caz': as~,'+bsY'+vsz'

=a1x'+aty'+asz': blX'+b2y'+bs~': c1x'+c,y'+osre'


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2 5 6 THEORY OF PLANE CURVESHence, the transformation formuloo reduce to the

forms- ~=ax+hy+gz'YJ=hx+by+fz'=gx+fy+cz

This shows that the point and the line are associated witheach other as pole and polar with regard to the general conic

I'.tXa +bya +cz!l +2fyz+2gzm+2hxy=0Thus it is seen that in case of reciprocals with regard to

a conic, the same line corresponds to a point, whether thatpoint be considered as belonging to the first or the secondsystem.

2 05 . POLE AND POLAR CONICS :The case of a point lying on its corresponding line ISinteresting and deserves consideration.Since a point (x, y, z ) lies on its corresponding line

~1;+17Y+'z=O, the locus of such points is obviously-(atx+b1y+C1Z),C+ (a!lx+bsY+c2z)y

+ (asx+bsY+csz)z=O

(1)and this is th s same conic, whether the point be consideredas belonging to the first or to the second system, and iscaned the "Pole Gonic."On the other hand, the envelope of lines which pass

through their corresponding points is a conic called thePolar Gonic.The co-ordinates of the point are expressed in terms of

the co-ordinates of their corresponding lines by the


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' ! " ' "

RATIONAL l'RANSFOIDIA'fIOXS 2 5 7equations (2) of 204. Therefore the required envelope is

(A.t+.A 2 17+A s ,) t+ (B.t+ B , 1 7+Bs ' )17+ (C .t+ C, 1 7+Cs ' ) '=O

where Au B 11 C 1> etc., have the significance as in 204.Conversely, the same pole and polar conics will be

obtained, if the points of the second system correspond tothe lines of the first system.

The pole and polar conics have double contact, theintersection of the common tangents being the pointCbs-c.), (c 1-as), (a.-b.). The chord of contact is foundto be the line (Bs-C.), (C. -As), (A.-B,),

It will be seen that the pole and polar conics areidentical, if b1= a . , bg = c . and C1=as. '* '206. QUDRIC I~VERsION :The process of circular inversion has already been

described in 15 ; but in this section will be described amore general process in which a point corresponds to a.point, while a line, in general, corresponds to a conic. Thistransformation can easily be effected by a geometricalconstruction and was given by Dr. Hirst.t In thisprocess a fixed point is taken as origin and a fixedfundamental conic as "base." Points collinear withthe origin and conjugate with respect to the base are said tobe inverse. If the base is a circle and the origin its centre,the points are ordinary inverse points with regard to thecircle. It' is, in fact, the circular inversion generalised andis called Quadric Inversion,

JI Scott-loco eit., 256.t Hirst-" a ll the Quad"ie [HL'e1'sio" of Plane o


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258 THEORY OF PLANE CURVES

Let C be the fixed origin and S the base-conic, ThroughC draw a transversalcutting the base in thepoints Q, R. Then, if P,P' are points on the trans-versal, such that (PP"QR) --is harmonic, then P and P'are inverse points,

Thus, to determine theinverse of a point P, wehave to find the point P',where CP intersects the polar line of P with regard toS, It follows hence that to any position of P corresponds asingle definite position of P', and vice 1'PJ'sa.

If P traces out a locus ~, P' will trace out a locus ~', and~' is said to be derived from ~ by quadric inversion.

R

\\

207. ANALYTICAL TREATMENT:Let CA and CB be the tangents to the base, and choose

4BC ,a.'lthe triangle of reference. Then the equation of thebase-conic may be written as-

(1)Let C r , y, z)P' respectively. and (,e', y', z') be the co-ordinates of P andNow, the polar line of P' is-

xy'+y,l;'-2zz'=0 (2)

whence,/'y'-fJa'y=O

;t; : y : z=x' : y ' : x'y'/z'(3)

and the line CP' is

=x'z': y'z': a;'y'= . ! . . . : . ! . . : ~ . . (4)y ' ;);' z ' These formulas are deduced on the supposition that the base is 1\

proper conic and the points A, B, C are distinct. Modifications arenecesaary when the base is a degenerate conic, and two or more ofA, B, C are coincident.


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RATIONAL TRANSFORMATIONS 2 5 9Iff (.r, y, z)=o is the locus of P, the locus of P' IS givenby the equation-

1 ( 1 " ~ " ; ) = 0y x zApplying the linear transformation x': y ' : a '=y' : x' : z '

i.e., interchanging the vertices A and B of the triangle, wemay express the result (4) in a more symmetrical form,and the locus of P' is now given by-

1 ( . ! . , - . ! . , . ! . ) = 0: 1 / y' Z '

The formuloo of transformation can, however, be writtenunder the form of bilinear relations=-

xie'=l, yy'=l, z .:; '=1 .2 0 8. QUADRIC h;VERSION AS RA'l'IOKAL TRA."NSFORMAl'ION.:Let the formulee of transformation in 199 be put into

the form-x': y ': .:;'=1 , : 1 . : / 3

where I" I II' 1 3 are rational functions of the second degreein e y, e.To the lines x'=O, y'=O, z'=O will then correspond the

three conics 1,=0,1 .=0,13 = 0 ; and in general, to a curveof erder n corresponds one of order 2n, obtained by puttingIIII i > I s respectively for x, y, z in the equation of the n-ic.

The simplest case presents itself in the form-x': y ' : z'=x' : y' : ~'.

To the line l.c+ lI~y+nz=O corresponds the conicl.v~+1n .vY+n .v~=0 inscribed in the triangle xyz. Similarly,to a conic there corresponds a curve of the fourth order, andso on.It is to be noticed, however, that this transformation is


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2 6 0 1'HI!lOltY o~ PLA.NE CURVESexpressed rationally in terms of (x, y , z ) , the latter are givenin terms of : - e ' , y', z ' by the equations-i, _i. _t,

r e ' - 11 - ;;-which are 110t rational and represent comes having fourcommon points; and consequently, corresponding to anyposition of ( . :1 ) ' , y ' , z ') there are foul' positions of (x , y, z )

But if illI..f., have one common point, since itis independent of the position of ( . - e ' , y', z'), it may beignored, and to any position of (x', y', z') there willcorrespond only three points (x, y , z). Similacly, if t.. f., f.have two points common, to any position of (e', y', z') willcorrespond only two positions of ( ,r, y, z ) . Finally, if i..f., fshave three common points, the conics have, besides the threecommon points, only one other common point, and to anyposition of (x', y', z') there corresponds only a single positionof ( .ll, y, z), and vice versa, and the transformation is birational.

Since it is perfectly legitimate to take three conicsof the form li,+mf. +vi, instead of t.. t.. fs' the threeline-pairs joining each of the three common points to theother two may be taken for L. f., fs' and the formulrobecome: ,c:y:z=y'z ':z ',J J ': x'y 'and x'; y': z'=yz : zx : xy.Hence, the quadric inversion is only a particular case of thegeneral birational transformation.

Other special cases may arise from the coincidence oftwo or more of the common points.Thus, when two points coincide, we may take the COIll'

mon tangent as the side y=O and the point (z , x) as thethird common point. 'I'he equations ofi.,f., fs will be ofthe form ax' +2fy.;+2hxy=O.Taking x', yz , ~ ;y as the three conics, the formulee become

x' : y' : z ' =.cy : x' : yzx : y : Z=,1;' y' : x" : 'J'z 'n d


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RATIONAL TIUNSFOR~L\'TIONS 26tSimilarly, when the three points coincide, the equabiousof ill t; is will be of the form-

by' +2h,fy+2i(yz-m~' )=0Hence, taking y', iVY,Y:-lnx' for i" t..I, respectively, theformulae become:

:/: y': z'=,ry : y' : yz-m,v'and il~: y : z=:e'y' : y " : y'zl-m;l;'2209. It has been stated ill 206 that the inverse of a

point P is a sillgle definite point P'. But there are excep,tional positions of P for which the inverse point P' is notin general determinate. The inverse P' is iudeterminate, if

(i) P is at C, P' is any point on AB,(ii) P is at B, P' is any point on BC,

(iii) P is at A, P' is any point on CA,(iv) P is any point on AB, P' is at C,(v ) P is any point on BC, P' is at B,

(vi) P is any point on CA, pi is at A.Hence it appears that if P is at any vertex or on any sideof ABC, the ordinary laws of correspondence do not apply.

210. THE bHRSE OF A STl(AI({UT LI:iE :'I'he inverse of the straight line l,v+my+nz=O (1)

i" the locus defined by the equation ( 207)l/y+m/x+n/z=O (2 )

which evidently represents It conic circurnsci-ibing thetriangle ABC.The following special case" are to ue noted :(i) If the line (1) passes through C, n=O and it IS its

own inverse,(ii) If the line passes through A, l::::::Ond the inverse

is the line 'mz+nx=O, which passes through B.


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2 6 2 THEORY OF PLANE CURVES

(iii) If thel.c+nz=O, andA .

line passes through B, its equation isthe inverse is the line h+ny=O throughThus, it is seen that the inverse of a right line is, in

general, a conic through A, B, 0; but in special cases it isa right line.

2 1 1 . PROPER INVERSE:Let 0 be the pole of a line meeting the base-conic in Q

and R. Then the inverse of the line is the conic ABOOQR.But if the line passes through C, then the pole 0 lies onAB, and the conic has three points on AB, i.e., the conicconsists of AB and the given line OQR. The line ABpresents here as a part of the inverse simply because theinverse of 0 is indeterminate, being any point on AB.When the line passes through A or B and meets the base-

conic in another point K, the inverse is a degenerate coniccomposed of OA, BK or OB, AK; for the pole 0 is now onCA or OB.Hence the points 0, A, 0 on the conic are accounted for

by the line CA or OB and the remaining points B or A andK give the other line.Similarly, if a curve passes through A (or B), the line OA

(or OB) presents itself as part of the inverse. These factors,however, occurring in the inverse are not regarded as formingthe proper inverse and are rejected. The remaining factorgives the proper inverse.

E, Consider the conic fyz + gz.;;+hilly =0The inverse of this, by the formnlre of 208, is-

f""'y'z' +gw'y"z ' + hw'y'z" =0l.e ., ./y'z'U~' +gy' +hz')=Oi.e., the sides AB, BC, CA and another line. Hence, rejecting the factorx"y 'z ', the propel' inverse is the line fill + g-y + h~O.


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. . .

212. THF. I:;"EllsE Of' THE LTXE AT hrIXITY:The equation of the line at infinity being a:l+by+cz=O,

its inverse, by the formulte of 207, is the conic-az.c+byz+wy=O (1)

which is evidently a conic circumseribiug the fundamentaltriangle.

The pole 0 now becomes the centre of the base-conic,the points Q, R ( 211) are the points at infinity on thasame. The polar of the point 0 at infinity on AB passe;;;through 0 and the line 00 is parallel to AB. 'I'here-fore the inverse of 0 is consecutive to 0 on the line 00, orin other words, the tangent to the inverse (1) at 0 isparallel to AB. Thus 00 is the diameter conjugate to ABand the tangent at 0 is parallel to AB. It may be noticedfurther that if the line drawn through A parallel to OBmeets the base-conic III K, BK is the tangent at B.

Similarly, the tangent at A may be constructed. The inverseto the line at infinity is represented in the figure by thedotted Iine.

213. INVERSIOX OF SPECIAL POIXTS OX A CCRYELet ~ be the curve and ~' its Inverse. The following

special points are to be noticed:If ~ meets AB in P, ~' touches OP at O. For, the inverse

ofP is on OP by definition, and as the polar of P passesthrough C , the inverse is indefinitely near to C on C Po


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THEORY 01


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RA'l'01NAL TRANSFOIOrATIONS 265Thus, if an n-ic ~ cuts OA (or OB) in n points, there is

an n-ple point on ~' at A (or B). When ~ touches OA (orOB), the two tangents to : S ' at A (or B) coincide, andconsequently, A (or B) is a cusp on ~'.

214. EFE'ECTR OF INVEHRTON O:-! SINGULAIlITIES :Prom what has been said above, it follows that, in general,

an ordinary point inverts into an ordinary point. But ifthree consecutive points at the given point and the threefundamental points A, B, 0 lie on a conic, their inversesare collinear on the inverse curve, and there is, therefore,an inflexion on the mverse. Thus the inverse of anordinary point may be either an ordinary point or aninflexion. Similarly, an inflexion is inverted into an ordinarypoint, unless the inflexional tangent passes through any ofthe fundamental points.Again, a double point, in genoral, inverts into a doublepoint of the same nature ; and consecutive double points invert

into consecutive double points, but the appearance may beslightly altered. Thus, a tacnode inverts into a tacnode, anoscnode inverts into an oscnode, but if the three nodesare initially collinear, the oscnode on inversion loses thisproperty, unless the tangent passes through a funda-mental point. Similarly, a curved oscnode may be straighton inversion.In the case of a bitangent, the inverse becomes a COIlIC

having double contact with the inverse, unless the bitangentpasses through a fundamental point, and then it inverts intoa bitangent. Conversely, a bitangent may be gained onmveraion.Thus it follows that as regards points and lines,

not belonging to the fundamental triangle, the pointsingularities of a curve and its inverse are the same, butline singulari ties are changed. Hence inversion canconveniently be used for analysing singularities on curves.


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14

266 THEORY OJ,' PLANE CURVES215. E~'FECTS OF INVERSION 01'1 A CURVE:*Let the curve ~ be an n-ic having a q-plepoint at A, an

r-pie point at B and an s-ple point at C. Then, ~ meets AB,BC, CA respectively at n-q-1', n-j'-s, and n-q-s otherpoints. The inverse ~' has therefore an (n-q-r)-ple,(n-1'-s)-ple and (n-q-s)-ple point respectively atC, A, B.Again, the q intersections of the tangents at A to ~ with

BC are points on ~'. Similarly,~' meets CA and AB in rand s points respectively other than A, B, C.

Since ~' meets AB in {(n-r-s)+(n-q-s)+s}i.e., (2n-q-1'-S) points, ~' is of order (2n-q-r-s).Thus, the inverse of an n-ic ~ with a q-ple point at

A, an r-ple point at B and an s-plepoint at C is a curve ~',of order (2n-q-r-s), with an (n-r-s)-ple pointat A, an (n-q-s)-ple point at B and an (n-q-1')-plepoint at C.

Putting n'='2n-q-1'-S, q'=n-1'-s,s'=n-q-1',

we may establish a reciprocal relation between ~ and ~'.Thus, n=2n'-q'-r'-s' q=n'-1,1-s'

r=n'-g'-8' s=n'-g'-r'We shall now show that, in general, the deficiencies of

the two curves ~ and ~' are the same.Since a q-ple point is equivalent to %q(q-l) nodes,

the deficiency P of the first curve ~ is given by-p=H (n-l)(n-2) -q(I}-I) -;{r-l) -s(s-I)}

Effects of inversion on higher singular points will be fnllydiscuased in Chap. XIII.


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RATIONAL TRANSFORMATIONS 2 67and p'=H(n'-I)(1t'-2)-q'(q'-I)-1"(T'-1) -s'(s'-I)}

=H(2n-'l- r-s-l)(21l-'1-r-s-2)- (n-r-s)(n- r-s-l)-(n-'1-s)(n-q-s-l)

-(n-'1-r)(n-'1-r-l) }=H(n-l)(n-2) -'1(q-1) -r(r-l) -8(8-1)}= r -

i.e., the deficiency of a curve lS unaltered by quadrictransformation.

216. ApPLICATION OP QuADRIC INVERSION:'I'ha process of quadric inversion affords a very convenient

method of investigating the properties of one curve fromknown properties of another. The following examples willillustrate the method.

E1). 1. Oonsider a conic cutting the sides of the fundamentaltriangle in three pairs of points.

Let a"" + by ' + cz ' + 2Jyz + 2gz ;!\ + 2h" ,y=O (1)be the equation of the conic cutting the sides BO, OA and AB in thepairs of points A" A, ; Bll B, and 0,. 0, respectively.

The inverse of (1) is the quartic curve-al? + bjy ' + c /z ' + 2f/yz + 2g!o> ; + 2h/? '!7J=O

The points A, B, 0 are evidently nodes ( 213) on the curve withthe lines All.." AA, ; BB" BB, and 00" 00, as nodal tangents.

But these lines all touch one and the same conic, and they areinverted into themselves. Hence we have the theorem :-

The nod"l tangents o J It irinodal. quavtic touch. on~ and the sameconic,

Again, the pairs of tangents drawn from A, B, 0 to the conic are alsoinverted into themselves, and their inverses are tangents to the trinodalquartic. :. Hence, the s;? ! tangents drmuH [ron: the three nodes to a trinodalquartic touch one and the sume e011 ic .

Finally, the' four bitangents of a trinodal qnartic are obtained by thesa-me process from the fact that through three given points, there call


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2 6 8 THEORY OF l'L.-\NE CURVESbe drawn four conics having doable contact with a given One, whilethe inflexional tangents are obtained from thc fact that through threegiven points can he drawn six conics, having three-pointic contact witha given conic.

E, 2 Show that the three cuspidal tangents of a bricuspidalquartic are concurrent.

If in Ell!. I, the pairs of points Au A,; B" E,; C" C, arecoincident, then the lines joining the vertices to the points of contactof the inscribed conic with the opposite sides are concurrent. Theinverse of the conic is evidently a triouspidal quartic, having the cuapsat the vertices, and the joining lines are inverted into themselves,which are again the cuspidal tangents, whence the truth of the theoremfollows.

E, 3. Through any point call be drawn two lines touching atrinodal quartic and passing through its nodes.This follows immediately from the fact that from any point only

two tangents can be drawn to any couic. On inversion the conicbecomes the trinodal quartic and the two tangents invert into twoconics through the nodes touchiug the quartic, and these evidently passthrough the inverse of the given point.

217 . CIRCULAR hiVEHSION : *A particular case of quadric inversion IS the trans-

formation by reciprocal radii the principles of whichhave been explained in 15. If we take k=1, the relationsbetween the rectangular co-ordinates of P and P' are-

X'= _ ! J _ .y'= a;2+y"and :: 1 "~-u=----,,e '2 +y"whence a / +iy'= ;lJ-iy'1

Writing X:Y 7,=.c-iy ; x+iy : 1and X' : Y' : 7,' =.l"+iy' : a!-iy': 1

< Moutard-Sn,. Z" truusfornuuion. P " " myuns vee/en!"s "eeiproques-Nouv. Ann. t; 3(2) (1864), pp. 306-30'J.


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RATiOS A L TItAXSFORMATIONS 2 6 9we obtain the relations X': Y' : ZI= YZ : ZX : XY,or in other words, the transformation is a quadric inversion.

The geometrical significnnce of these transformationswill be best understood from the figure of 206, if weconsider that the points A and B are circular points at infinity,so that the base-conic S now becomes a circle with centre Cand P, pI are in verse points with respect to the circle. Infact, we have taken the circular lines through the origin andthe line at infinity as the sidos of the triangle of reference.Hence, circular inversion is a particular case of quadrictransformation, and quardric transformation is ageneralisation by projection of the process of inversion

vVe may deduce a number of theorems from the resultsestablished in the preceding articles. Thus, the inverse of acircle is a circle, that of u straight line is a circle throughC, and so on.

The inverse of a conic, in general, is a trinodal quartic,the nodes being the origin (C) and the circular points atinfinity. If the origin be the focus of the conic, the inverseis a limacon , if the origin be on the conic, the inverse is anodal circular cubic, the origin being the node.

An osculating circle to a curve will invert into anosculating circle of the inverse, but when the circle passesthrough thc origin, the inverse is an inflexional tangent.

218. Spr;CIAL QUADRIC TIIANSFORJIATlOIlS:In 206 we have discussed the general case of quadric

transform/ltioll; but special cases arismg from specialpositions of the points A, B, C or the nature of theBase must he considered for a systematic treatment of thesubject.

Case 1: One special case presents itself when the pointsA and B coincide ( 208.)


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2 70 THliORY 01" PLANE CURVESIII this case, any line through C, the line CA, and the

polar of C are taken as thesides of the triangle ofreference. The base-conicIS now a pair of lines,whose equation, by aproper choice of co-ordinates, can be put as;t-z"=O. The polar ofany point P'(x',y',o') is theline X.l/-zz'=O and CP' is ~ty'-x'y=O, whence the inversepoint P is given by-

A

c

z : y : z=:J.l' : y' : f"/z'=,lJ'Z' : y'z' : : 1 : "~' : y' : z'=za; : yz : a;2=.c : y : x'iz.

Hence, if l(x, y, z)=O be the locus of P, that of P' IS

It is to be noticed that this transformation is equivalentto the three transformations in succession, in which the poleis the point C(O, 0, 1) and the bases are the three conics

x"-xy+z'=O, m'-y'+z2=0 and x+xY-Z2=0.Case II: When the three points A , B , C coincide at C ,

any chord through C , the tangent at C and the tangent atthe other extremity of the chord are taken as the sides ofthe triangle of reference.The base-conic is now of the form 2yz-mx' =0, where

1n is at our disposal.The polar of P'(.()', y', z') is y'z+yz'-ma:x'=O and CP'

is the line xy'-x'y=O, whence pea', y, z ) is given by-

andx : y : ~= ..c'y' : y" : m,lJ"-y'z'

x' : y' : ,.'=:cy : y2 : mx' -yz ( 208)


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RATIONAL 'l'RANSFORMATIONR 2 7 1

219. KOETHER'S TRANSFORMATION:We have so long used the same triangle of reference for

the our're and its transform; but if we take CBA instead ofABC as the triangle of reference for the transformed curve,this amounts simply to the interchange of IV and z in thetransformed equation. Hence, the curve f(x, y, z)=O istransformed into f(z, y, z/.c)=O

Writing this equation in the form f(x/z, a y / z 2 , 1)=0we see that in the Cartesian system, the curve f (:c, y)=O IStransformed into f(x, 'l'Y ) =0.

Hence, the ormulre of transformation become-x : y : 1=:t' : x' y' : 1, i.e., a=.c', Y='JJ 'Y ' and y '=yj.l' , a /=,r.'I'his form of transformation was given by Noether * and

was used by Newton and Cramer for the analysis of highersingularities. A series of successi vp transformations are attimes required for complete analysis.

E :D . 1. Examine the singularity at 0 on the curve y 'z=w '.The inverse by the forrnuleo is x"y"=lJ" , and consequently

the proper inverse is y " = a;'2,which has at 0' (x', '!I') a cuspwith JII' =0 for tangent.

Oonsequently, the singularityat 0 on the original curve is atriple point ( 213) whoseapparent form is that of aninflexion, but the penultimateform is shown as in the figure.

E: 2. Verify the following:(i) The inverse of a line through C or B is a line through C or B .(i. .:) The inverse of a line is a conic through 0 touching AB at A.E z, 3. Examine the singnlarity at the origin on the curve y' = ,,,'. Noether-Uber die siugularen Wertsysteme einer algebraischsn

Function und die singularen Pnnktc einer algebraischen Curve-e-MathAnn. Bd 9 (1876), pp. 166.182.


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-272 THEORY 01" PLAN8 CURVE:;

220. CrmMONA CONDITIONSAs explained in 199, the general transformntion

;r/: y ': z '= I, : I . : 1 3 is not birational, i.e., from thissystem it is not, in general, possible to deduce another ofthe form x : y : z=F/, : F'. : F', where F'" F/2l 1 !", arerational functions (polynomials) in ,r/, y', z ',

Luigi Cromona * has investigated the conditions underwhich such mutual expressions are possible.If in the one system we are given ;1;' : y' : zl=a : b : c,

then the corresponding points in the other are given as t.heintersections of the curves-

( 1 )Now, smce i.. t.,/3 are polynomials of the nth degree

in (x, y, z), the number of intersections will be n2. Butif 1 1 ' t.,1 3 intersect in p common points, the curves (1)will evidently pass tbrough these common points, and theremaining n" -ppoints will then correspond to the givenpoint (a, b , c).

When p =n" -1, the curves will intersect only in onevariable point, that is to say, all but one intersections of thecurves (1) being known, the co-ordinates of the only remain-ing point will be determinate, and thus rational functionsof (a, b, c.), i.e. of ,t/, y ', z', and we shall have- ,

x : y : z= F / , : F'" : F / 3Hence we see that this will be a birational transforma-

tion, if the three curves I" 1 . , 1 3 bave n2-1ommon pointsof intersection.

" Cremona has thoroughly investigated these conditions and thetheory is due to him-see his Memoir Bulle transjorma zioue qeometricliedellefig'u"e piane-Mem. di Bologna, Vol. II (1863). and Vol. V (1865),For applications of Oremona transformations, see a paper by A. B.

Coble in the Bull. of the Am. Math, Soc., VoL 28 (1922), pp. 329-364,to which is appended a number of important references on the subject.


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RATIONAL TIUNSFORMATLONS 2 7 3This agam IS not :t sufficient cond ition. For, ifi..i; i, be cubic curves having' eight common points,

they certainly have a ninth point common, and consequentlythere is no variable intersection. But here again, if wesuppose that the enhies have on e node common to all, theyintersect in four other ordinary points, and since to begiven a node is equivalent to three conditions, seven of theirintersections are known, and therefore, only two more con-ditions are required to determine any curve afl +bf2 +cfs =0Now, the common points are equivalent to eight intersections,the node counting' as fonr. Hence, one variable point isobtained corresponding' to the giyen point.

In fact, the system of curves afl+bf2+C!3=Ocorres-ponds to the system of lines a,t!+ by ' +c:' =0, * and shouldtherefore be perfectly general and must not be determinateexcept when a, : b : c are given, which is equivalent to twoconditions.

Therefore, the number of conditions to be satisfied byi..f., i, must be at least two less than the number of con-ditions determining a curve of order n.

221. From the above considerations it follows then thatn being greater than two, i.. f2,f3 cannot have n2-1common points, for then they have another common pointand no variable point of intersection. If, however, t..f., f ahave U1 ordinary points, U2 double points, U3 triple points,etc., common, such that these are equivalent to n2-1intersections and the number of conditions thus implied bitless by 2 than the number necessary to determine a curveof order n, we obtain on" remaining variable point of inter-section corresponding to the given point and the transforma-tion becomes rational.

Since, to be given an r-ple point on a curve is equivalentto {1' (r+l) conditions and two n-/c8 intersect in r2 points

See Montesano =Napoli Rendi, Vol. 11(:1) (1905), p. 25!l.


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274 THEORY OF PLANE CURVESat an I 'p le point on each, we may state the above twoconditions as follow:

a, + 2 2 . : > . , +3'a. + . . . +12a,.=n2-1 (1)and a, +3a.+6a.+ ... +-}?(?+1)ar=tn(n+3)-2 (2)Combining (1) and (2), we may state the second conditionIII a simpler form :

a, +2a. +3a. + +ra, =3(n-1) (2')Positive integral values of at> a., ... satisfying the

equations (1) and (2') will then determine the transfer-mations, provided the number of higher singularitiesassumed to belong to the curves does not exceed the properlimit. Cremona has tabulated all the admissible solutions,for cases up to n=10, of the above equations, which areoften referred to as "Cremona conditions."For a detailed discussion of the theory, the student.

IS referred to Cremona's Memoir and to Cayley's paperabove referred to and also to his paper--" Note on the theoryof the rational transformation between two planes and ofspecial system of points, Coil. Works, Vol. VII, pp. 253-55.222. THEOREM: EveTY Cremona tmnsformation may be

reduced to a number of successive quad?ic tmnsfm'mation;* andconverselu, each birationai transformation of a plane into another~8 equivalent to a finite number of quadric transformations,

Consider the transformation:x' : y ' : =t, :f. : f3 '

where f" f., f. are curves (polynomials) of order n,having a, ordinary points, a. double points, etc., in common.

Vide Prof. Cayley's paper-" On the Rational Transformationbetween two spaces," Coil. Works, Vol. VII, pp. 189.240.

For other proofs see Noet.her, Ueber Flachen etc., Math. Ann. Bd.3( 1871), pp. 161.227, Segre, Un'osservazione relativa, etc., Torino Atti,Vol. 36 (1901), pp. 645651 and Castelnuovo, " Le transformazioni,eto.," ibid, Vol. 36 (1901), pp.861.874.


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RATIONAL TRANSFORMATIONS 275Then, there are three of these points '* (one q -ple, one r-pleand one s-ple, say) the sum of whose orders exceeds n, so that

Now, take those three points as principal points of aquadric transformation. Then the degree of the transformedcurve is, by 215, 2n-q-,-s, which is certainly less than11., i.e., the degree of the given curve is reduced, and by asecond quadric transformation, the degree of this curvemay be further reduced. Proceeding in this way, we shallultimately obtain rig'ht lines corresponding to the n-ics.'rhus the Cremona transformation is reduced to a numberof successive quadric inversions.

But, it was proved that the deficiency remains unalteredby quadric transformation, and consequently it remainsunaltered by any Cremona transformation.

223. DEFIC I&NCY U)lAllmRIW 13YCR";.\IOXA TH.AN:;r'ORMATJON :Let F=0 be a curve of order l c and let us apply the

transformation to this curve. If now IIII., I, have a pointA in common, the line corresponding' to A will meet thetransform in k points all corresponding to A, which thenbecomes a k-ple point. In general, any of the r-plepoints becomes a kr-ple point. Hence, if the given curve hasno multiple points, the transform will have none except atthe principal points, i.e., at the common points of 1 1 ' f. and is.

Thus, the degree of the transform is nk and the corres-ponding' maximum number of double points, as usual, isHnk-l)(nk-2). Also the multiple points at the principalpoints are equivalent to-

~alk(k-l)+!a a '2k(2k-l)+ ... +~a,.h(rk-l)or


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2 7 G THEORY OF PLANE CURVES

which, by equations (1) and (2') of 221, is equal to~k'(n2 -1)-~k.3(n-l) =tk2 (n'-1) -ik(n-l).

Hence, the deficiencyof the transform becomeR-Hnk-l)(nk-2)- {~k' (n2 -1)-~k(n-l)}

=Hk-l)(k-2),the same as that of the original cnrve.If, however, the original curve has other multiple

points, the transform will have corresponding multiplepoints of the same order and the deficiency will remainunaltered ( 222). Further modification is necessary whenthe original curve passes through any of the principalpoints.Again, when the curve F=O passes through the principal

points au a., ... the degree of the transform will be-N=:nk-u,-2u2-3a3 -1'(1,.

224.Rn:~IANN TRANSFOIDIA'l'ION:Wehave hitherto considered the Cremona transformations

which are birational with regard to points ofthe whole plane,under certain conditions. But there are other transforma-tions that are birational * only as regards the points of acurve of the plane, but no such conditions are uecessary illthis case.Let F=O be a given curve and apply the transformation

where t, 1 2 , fa are homogeneous functions of the nth degreein ";,y, z, not necessarily satisfying Cremona's conditions,which have no common factor. The above equations arenot by themselves sufficient to express .e,y, z rationally

* For the birational transformation of a curve into itself, see H. A.Schwarz, Crelle, Bd. 87 (1875), p. 189, also F. Klein, Uber Riemann'sTheorie del' atgebraischeu Functionen (1882), p. 64.


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RATIONAL TRANS1 c p '. , c p 's are homogeneous functions 111 x ' , y', zof the same degree 'It, without a common factor.

In fact, when x, y, z are eliminated between the equationsof transformation and F=O, we obtain an equation l!"=O,which is the condition for the co-existence of the system ofequations, When this condition is satisfied, z, ?I,;; can bedetermined rationally in terms of x', y', " , ' , *

DEFIN[1'ION: An algebraicbirational as regards the pointsregards t.he points of the wholeTransjormatiou,

transformation that ISof two curves but not asplane is called a Riemann

E J;. 1. Consider the two curves-z (y ' +w ')=x' and Z'2y '=W'3

both of which have the deficiency zero. We can determine a trans-formation which will transform the two curves one into the other.

Any point on the first can be expressed as-ill : y : z= (1 +A') : A(1+A')

and any point on the second is given by-~' : y' : z'=l\.'z : J\'3 : 1.

If now we associate the points of the two curves which hare thesame parameter, i,e., A=A', then

1 L y'W =;1 and . : . = _1_ 1+ A",z

whence x : ' lJ : Z= ~/(a.:'+ z') : y'(.v' + z'} : z'x',and also ill' : u ' : z '= x(J J -z ) : Y (J J -z ) : z .If, again, A and A' are connected by a bilinear relation of the form

AAA'+ BA+ CA' + D=O, we may, in a similar manner, express x' : Y' : z 'in terms of x, y, z.

Ii Salmon's Higher Algebra, Lesson X.


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2 7 8 THEORY OF PLANE CURVESEx. 2. Consider the two curves-

andIV : y : z= t' :t: 1+ (2X' : y ' : z ' =(t 2 -1) : (t 2 - 1)' : t.

Associating the points which have the same parameter, we shallshow that :Il, y, Z can be expressed rationally in terms of :1 ) ', s', 2 ', andvice t'ersa.

In tho first curve,x

and. , , "

(1)=tY z 1+ t'In the second curve, : v ' z' and x' =t'-1 (2)=t . , ' ; 1' X

and xzF ~'+z'1 + t' - .v ' + 2z '

whence y z=1 y'z'W '2 a/ +2z'. 1 : ' +:,'=x"(.,' +2/) : y 'z '(x' +Z') : .v ',(., ' + 2~'). (A)

II' z'~ =t.-y' QJ'

and

002_y1 y":.-al': y ': z'=I: ----- : --. =xy(x'-y ') : (x'-y ') ': xy '. (B)xy .V'_y2'I'hus, by Riemann transformation the two given curves can he trans.formed one into tbe other.

Ex. 3. Consider the curve X'+y3+Z '=OApply the transformation w': y' : z ' =x' : y' : z'.

Now e : y: Z=2X+y3 z3 : 2x3Y 'Z3 : 2Z3y 3z '=x+(x" +2y3z3_X6) : Y '(Y " +2Z3X3_y6)

: Z+(Z6 +2X3y3_Z6)=z '{x. +2y3 z3_(y3 +Z3)"} : y 'fy" +2Z3X3_(Z ' +a;3 ) '}

: zO fz " + 2x3Y "_(X3 +y3) '}=x+{2Z"-(x" +y" +z " )} : y+{2y6_(X +y. +z " )

: z'{2z0-{x6 +y. +Z6)}=" {2 .C '3 -k} : y" {2 y'3 -k} : Zl. {2Z '3 -k}

where


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ItATIONAL TRANS}'ORMATIONS 279Thus, z, y, z have been expressed rationally in terms of J:/, 1/', z' withthe help of the equation of the given curve,

Now applying this transformation to the given curve, we have forthe equation of the transformed, after rejecting a factor,

which gives the reciprocal polar cnrve of x3 + y3 + Z3 =0, and as isknown, there is (1, 1) correspondence between the points and linesof two reciprocal figures,

225. REDUCTION OF THE ORDF.R OF THE 'l'RANSFORMF.llCURVE: '* '

From what has been said before, it follows that if weapply the transformation of 221 to the n-ic F, the orderof the transformed curve will be Nsnk-a, -2a ... etc.,where a ll a., etc., denote the number of single, double, etc.,points common to the k-ice III 121i, lying on F. We shallnow consider how this transformation can be applied so asto reduce the order of the transformed curve as low aspossible, i.e., to make N a minimum.Now, the curves 11 'I.,Is can be made to satisfy, as has

been seen in 221, ik(l,+3)-2 conditions.Hence, N will be a minimum, if i ll i2 1 is be assumed to

pass through as many as possible of the double points of thegiven curve F.If then the deficiency of F be denoted by P, the number

of its double points is-Hn-l)(n-2)-p, i.e., 1n(n-:3\-p+1.

(i) Suppose k=n-l.Then, t..I.,may be made to pass through

1k(k+3)-2=-Hn-l)(n+2)-2=tn(n+l)-3 points only.

ot. Salmon, H. P. Curves, 365.


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:!80. THEOlty 01-' PLANE CURVESTherefore, besides the double points, the curves f l t.. t,can be made to pass only through

{}n(n+1)-3}- {tn(n-3)-p+l}i.e., 2n+p-4 ordinary points on F, so that we may take

a,=2n+p-4 and a.=tn(n-3)-p+l.Therefore, the order of the transformed curve is

=n(n-l)-(2n+p-4)-2{{n(n-!3)-p+ I}=p+2.

U'i) Put k=n-2, (n>2).Ail before, we may take a.=t12(n-3)-p+l; RO that

al =B-k(k+3)-2}-Hn(n -3)-p+l}=H(n-2)(12+ 1)-2}-Hn(n-3)-p+ I}=n+p-4N=n(n-2)-a, -2a.=12(n-2) -(n+;o-4) -2{tn( n-3) -p+ I}=p+2.

(iii) Put k=n-3.We take a. =tn(n+3)-p+l, and consequently, al =p-3

as before, so that p is to be taken always greater than 2.Hence, N=p+l.Since the transform has the same deficiency as the given

curve, we may summarise the above results in the followingtheorem:A curve of order n with deficiencyp may be transformedinto a curve of order p+2 with deficiencyp or with tp(p-l)double points.If P >2, the order of the transform may be p+1 with

deficiency p, or with -}p(l-' -3) double points.


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RATIONAL l'R.\NSFOR~L\TIONS 28]If, however, p=O, the curve may be transformed into

conic, which however can be further transformed into astraight line.If p=1, the transform is a cubi so on.For a detailed discussion, the student IS referred t.o

Brill-Noethers paper-s-" Ue11e1'die algebraichen Fnncktionenand ihre Anwcndung in del' Geometrie "-l\IatJl. Ann.Rd. 7, pp. 297-398, and also to Cayley's paper-" On tl letrnnsforrna.tion of plane curves," ColI.Works, Vol. G , pp. 1-9.

2 2G . REOUCTIOX OF A CURVE WITIl ~IULTIPr.F. PO[!(TS:The following formal proof for the general case of a CIlj'\'C with

mnlt.iple points was given by Scott.'Let F have mult.iplo points of orders ,." "",." enc., and at these

points let the curves jll 1 2 ' t, have multiple points of ordersP" POl p" .. etc. (whero nny of the ,.'s or p's may be zero or unity).

It is required to determine I.;and p's, so that the order of the trans.Io rmnd curve N=nk-~r~p~ may be a rnin im nm , i.c'lfora~ivcn\'all1f\of k; : : : " . P ; is to be made a maximum.

The cnrvesf"f,,/, can be made to satisfy 11.(k+:3)-2 conditionsOldy, but if a p.point of the's is placed at an ordinary point of F, thennmber of conditions imposed is ~p(p+ 1), while the point counts as pintersections.Evidently, '; ;p(p + 1)~p, according as p~l. Hence, all ordinary point

on F will count as most intersections, if it be an ordinary point ofj' , i.e., if p= 1.

Again, if a p-point is placed at an r.point, the number of conditionsis -Isp(p + I), while the number of intersections is "p, and I'P-~p(p + 1) iscertainly a positive quantity, if ,.>1 and p=l, and generally, thedifference betwe.on the number of intersection . and the number ofconditions is to be made a maximnm.

Since the multiple points are snpposed independent, the existenceof other multiple points will not affect the number of conditions imposednpon I" I" Is by supposing the p.point at the ,..plc point of F. Hence

Scott, "Note on Adjoint Curves," Quarterly Jonrnal of Math.,Vol. XXVIII, pp. 377381.


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: --_. . . .

2 8 2 THOORY OF PLANE CURVJ


35/38

lL\TION:\L THANSFORMATION~ 2 S : SNow, writing k=II-3+1, we have 2


36/38

Z84 T H J ; ; O R Y at' P L A N E C U R V E S

It IS to be noticed, however, that for n=1 or 2, p=O;and for n=3, 1'=0 or 1 according fLS the n-ic (cubic) hasor has not a double point,

228. IN'l'fo:RsEcT!o~s Ob' A CUBVE WITH lTS ADJOINT:Since at every r-ple point the adjoint has an (1'-1 )-ple

point, the point counts as 1'(1'-1) intersections, and thefact that the adjoint has an (1'-1)-ple point is equivalentto ~1'(r-1) relations between its co-efficients.

Hence we obtain the theorem:'1'he number of intersections of an n-'Ie and an arl; j( ) 'int at

the multiple points of the u-z 'c is double the number of rela-tions between the co-efJicients of the adjoint curt:e.If the adjoint be an (n-3)-ic, since there are p-1

arbitrary co-efficients, the Dum bel' of relations between itsco-efficients is -~n(n-3)-p+1=}(n-1)(n-2)-p,

t. Show that the identity (1) of :38 holds, if CIII, em " C/,' C / , 'are adjoints to C /I

229. IK'l'ERSECTlO~S 'YlTH A PEKC/1. OF ADJOIN'l'S :Let l: be the order of a curve adjoint to the n-ic, with

multiple points of orders '1'1,1'" 1'" .. , Then the multiplepoints count as l'r{1'-1) intersections and the co-efficientsof the adjoint 7,ic aie connected by r~'I'(1'-1) relations.Therefore the lr -ic requires tk(l,+2)-ilT(-r-1) ether

conditions to be uniquely determined, i.e., we may takei"( k+3)-t~r( 1'-1) other ordinary points on the u-ic besidesthe mul tiple points, so as to completely determine the adjoint.Now, the two curves intersect in nk points. Hence the

numbel' of remaining intersections=nk-~I'(1'-l)- Hk(k+;3)-i~I'(I'-1)}=nk-t~r()'-l) -~"k(k+3)=nk+p-H n-1)(n-2)-tk(k+:3)=H2nk-n2 -Ii' +8n-3k)+p-1=Hn-Z)(k-n +3)+p-l.


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RATIONAL TRANSFOR1IATIONS 285This result shows that if we describe a pencil of k-ics

through the multiple points and through{t7,(k+3) -I}-~:SI'(r-1)

other ordinary points on the n-ic, then this pencil willmeet the n-ic in -}(n-k)(k-n+3) +p variable points.Hence, we may state the theorem :Any curve of a pencil of adjoint k-ics, through the 1Iwltiple

points and other ordinary fixed points on the n-ic, will meet then-ic in tin-k)( k-n+8) +p variable points.If k=n-1 or n-2, this number is p+ 1; if k=n-3,it is equal to p.Thus, any adjoint (n-3)-ic through the multiple points

and through -in(n-8)-1--i-(n-1)(n-2)+p, i.e., p-2ordinary points on the u-ic will meet the n-ic in pothervariable points.

Ere . A poncil of adjoint k-ic has its base points on an n-ic , Showthat (n-k)(k-n + 3) + 4]1-2 curves of the pencil touch the n-ic atpoints other than a base-point.

230. TRAXS,'ORMATIOX BY AD.JOIKTS :Let there be ", double points, ", triple points, and a,. (J' + 1).ple

points on the given u-ic F = 0, so that the adjoint k-ics havecommon a, single points, ", double points, ... "-ple points on F.

If p' denotes the deficiency of the adjoints, by 63.p =}(n-l)(n-2)-}:J:I'(J' + 1)",.

p' =}(k-l )(1.--2) -~tl'("-1 )",Now, p=-}(n-l)(n-2)-}tl'(I'-I)",-(a, +2a, + ... +1'a,.)

=}(n-l)(n-2) -}tr(r-l)a, -3(7,-1). ( 221). 1 > ' -p= {,l-(k-l)(k-2)-}tr(r-l)a,}

- H(n-l)(n-2) -}tr(r-l)a,. -3(k-l)}

=}(k-l)(k-2) -&(n-l)(n-2) + 3(k-l)i,e. p' =~(k-l)(k-2) -}( n-l )(n-2) + 3(k-l) +p.

(1)(2)


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2 8 6 TUJ,,-3, we have IT=llk-p-2-~r(-i'+ l)a,. Substituting

this value of

Ganguli, Theory of Plane Curves, Chapter 10

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Transcript of Ganguli, Theory of Plane Curves, Chapter 10