Ganguli, Theory of Plane Curves, Chapter 2

8/14/2019 Ganguli, Theory of Plane Curves, Chapter 2

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CHAPTER IIPLANE ALGEBRAIC CURVES

Sec. L: - General Properties.17. NOTW" O J< ' A LG ED H AIC CURVES.If on the plane of the paper we draw a line of any

form, it is called a "curve." It is theoretically possiblealways to represent this curve, described according to certainlaws, by means of a definite analytical equation in anysystem of co-ordinates, or at least by means of a FourierSeries. We propose to investigate the properties of acurve represented by means of an equation F(./', y) =0 inCartesian co-ordinates, and in doing so, we shall have firstto study the nature of the function F(;e, y).

In former days, functions were divided into two separateclasses- (1) algebraic, and (2) transcendeutal, and thecurves represented by them were accordingly divided intotwo classes *-" algeb raic C 1t1'ves"and" transcendental curves,"This classification was chiefly based on the use of Cartesianco-ordinates and therefore does not hold when other systemsare used. But this is of little importance compared withthe great advantage we derive in studying the propertiesof algebraic curves in the light of the modern theory offunctions. For functions both of whose ranges are realnumbers, a graphical representation was devised byDescartes. The older Mathematicians held that a functionsimply meant a single formula, at first usually a power ofthe variable, but afterwards it was regarded as defined byany analytic.al expression, and was extended by Euler t toinclude the case in which the function is given implicitly

> I I ' Descal'tes-Geomet1'ie (Leyden, 1637).t Memoil'eH de I' Aead des Be T.4 arid 5.


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22 'fH~ORY m' ptAN~; CURVES

as a formal relation between the two variables. Thearbitrary nature of a function given by a graph wasreeognised by Fourier. In the modern theory of functions,it is held that a regular function can be completely definedby means of a graph drawn in the finite and continuousdomain of the independent variable. 'I'his simply dependsupon geometrical intuition. A curve thus drawn isindistinguishable by perception from a sufficiently greatnumber of discrete points. But a graph can only beregarded as an approximate representation of a function,and all that is really given by the graph consists ofmore or less arithmetically approximate values of theordinates at those points of the .e-axis at which we areable to measure them.

In order that a curve may be drawn really to definea function, certain laws must be formulated, by means ofwhich the values of the ordinates can be formally deter-mined at all points of the x.ax-ill. These considerationsled to the classification of functions in the modern theory,according as they possess various special properties, namely,continuity, differentiability, integrability, etc., throughout thedomain of the independent variable, or, at or near specialpoints in that domain. The modern theory of functionssays that the equation F C r , y)=O cannot in generalrepresent a curve; it can do so, if y can be expressed as aregular function f( ' 1 ' ) of . 1 ' , i.e., if fea;) is a continuous, finiteand unlimitedly differentiable function expansible inTaylor's series. It is only by the combination of theseconditions that y=f(,v) can represent a curve.

18. REPRESENTATION OF FUNCTIONS:Each algebraic function is put in the following rational

and integral form:-1n 1

Ft.!',!!) - = 2 ~ a" :l;'y'.~'=O r=O


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GENERAL PROPERTIES 23

All other functions which cannot be put into this form arecalled " transcendental" functions. A class of functionsof the form y=,c.v2 is called by Leibnitz "interscendental "functions. These functions involve variables with exponentsnot commensurable with any rational number.

Parametric representation of functions is often very usefulin the form x=cp(tl , y=tf!(t), from which the implicit formof the function is obtained by eliminating the variable t,

19. From the preceding articles it follows that undercertain conditions Fe," y) =0 is the equation of an algebraicCUrTe,while the curve itself is the geometric representationof the function Ft,!" y). Thus, in general, a curve of thenth order is defined as the geometric representation of Itfunction which is of the nth degree in the variables,Different special names have been devised to denote curvesof different orders; for instance, a curve of the third orderis called a "cubic," of the fourth order a "quartic," andin general, one of the nth degree is called an n-ic.

If the quantic F\", y ) is irreducible, i.e., if it, cannot bebroken into two or more rational and integral factors oflower dimensions, the curve is called a "propel' or non-deqene-rate curve I, of the nth order. But if F(",y ) breaks up intotwo or more rational and integral factors of lower orners,the curve is called an "impl'opel''' or "degenerate" curve.Thus, two right lines together form a degenerate conic j aconic and a line, or three right lines constitute a deqeneratecubic; two conics constitute a " degenerate quartic," and so on.

20. The most general equation of the nth degree in twovariables may be written as :-

a+b~:+('y

+d.c2 +e,vy+fy'+gcS +h.v"y+i..,y +n'+ .

+p.c"+q,t:"-ly+ "'+r,cy"-l +sy"=O (1)


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24 THEORY OF PLANE CURVESThe above equation will sometimes he written 111 the

symbolic formuo+ u,+ u.+ u.+ ... + 'It,.+ ... +'It.=O (2)

where Uo is a constant and u ; is a binary quantic (1'-ic) in,cand y.The equation (2) may be made homogeneous by intro-ducing a third variable z, and the general equation ofa curve of order n in homogeneous coordinates may bewritten as:

U OZ + 'lt ,Z - '+u.z- 2+ ... + U,Z -'+ ... + 'lt.=O .,. (3)From these equations it is seen that the number of

terms in 'ltr_, is 1', and that in u , is obviously one morethan in 'It''_l> and is therefore equal to 1'+1. Thus thetotal number of terms in each of these equations is equal to

1+2+3+ +n+(n+l) i.e., to Hn+l)(n+2).The number of independent constants in each of these

equations is equal to one less than the number of terms itcontains, for the generality of the equation remainsunchanged, and consequently it represents the same curve,if we divide the whole expression by a constant. Thus,dividing the equation by uo' and substituting new constantsfor the ratios of old ones to 'lt~, the number of constants isreduced by one. Hence the general equation of the nthdegree contains only H(n+l)(n+2)-1} ,)1' tn(n+3)independent constants and can therefore be made to satisfythe same number of conditions, and no more.

21. NU~IBf;;lt OF POINTS DETE1Df[NI,W A CURVR OF ORDER .

The general equation contains tn(n+:3) independentconstants and therefore the same number of conditions arerequired to determine the co-efficients uniquely. Hencetn(n+3) conditions are required to determine a curve oforder n; for instance, the curve may be made to passthrough tn(n+3) given points. Thus, a curve of order n


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GENERAL PROPERTIES 25is uniquely determined, if -}n(n+3) points on it are given.The co-ordinates of each of the given points satisfy theequation and thus give a linear relation between theconstants. We obtain in this way altogether -}n( n+3)equations to determine the same number of unknownquantities, i.e., the co-efficients. There isjonly one set ofsolutions, and the equation is uniquely determined, andconsequently the curve also. Hence it follows that

Through -}n(n+3) git'en points, only one curve of the nthdegree can be described.

Thus a conic is determined by five points, a cubic bynine points, a quartic by fourteen points, and so on.

22. DEGENERATE AND NON-DEGENERATE CURVES:We have seen that -}n( n +3) points will determine

a curve of order n uniquely j but they do not in all casesdetermine a proper or non-degenerate curve. For instance,five points determine a conic, but if three of them lie on aright line, the conic determined by them consists of two rightlines. Nine points determine a cubic, but if three ofthem lie on a right line, the cubic consists of this line anda. conic through the remaining six points. Thus we seethat the -}n(n+3) given points do not always determine anon-degenerate curve of the nth degree.

The necessary and sufficient condition that they candetermine a non-degenerate curve uniquely is that thepoints must all be "independent," i.e., no group of themshould lie on a curve of order lower than they can usuallydetermine. This follows from the fact that the linear relationswhich determine the co-efficients must all be independent,and their number must be exactly tn(n+3), neither more norless. For, if two or more of the co-efficients are connectedby other relations, the co-efficients are not independent, butthey satisfy identical relations. In this case some of thepoints lie on one or more other curves of lower orders and


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2(\ THEORY OF PLANE CU R.YFlSconsists of two or more curves of lower orders. If thenumber of these linear relations be less than -}n(n+3),which is the case when one or more of them can be deducedfrom the others by algebraic operations, the co-efficientscannot be uniquely determined and the equation containsone or more indeterminate co-efficients. In this case aninfinite number of curves can be described through thepoints.

Thus we see that in gene1'al-}n(n+3) arbitrary givenpoints determine a "non-degenerate curve," and thatuniquely.

23. lNTERSECTTO~S OF CURVES:Two cun;es of orders 'In and n respectively intersect in geneml

in mn points.Let

(1)(2)nd v. ::a', +b',;l:+c',y+d',x~ +...

be the equations of two curves of orders 'I1t and n respec-tively.

Then, at the common points of intersection of the twocurves both the equations (1) and (2) must be simulta-neously satisfied. Therefore, if we eliminate either of thevariables ;cor y between them, the resulting equation willbe satisfied by the values of the other variable at thecommon points of intersection. Thus, if y be eliminatedbetween (1) and (2), the resulting equation in a -theeliminant-will be of degree mn and will determine theabscissee of the points of intersection of the two curves.

The resulting equation will have either all its rootsreal, or if there are imaginary roots, they will occur in pairs.Therefore if there are 2k imaginary roots, the number of realintersections will be mn-2k, where k is zero or a positiveinteger.


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28 THEORY OF PLANE CUR~ES

mn-tm(m+3) points are thereby determined. If m~n,mn-Hn-I)(n-2) points can be assumed arbitrarily, andthen the remaining t(n-I)(n-21 points are determined.For m=n-l orl1~=n-2, the two numbers mn-tm(m+3)and tCn-l)(n-2) are the same.

Hence we can say that, when m~n or m=n-lor n-2,mn-t(n-l)(n-2) points can be chosen arbitrarily and theremaining Hn-I)(n-2) points are thereby determined.*

25. If m=n, the above theorem becomes-if of the n2points of intersection of two Ct~rvesof the nth deqree,n-t(n-l)(n-2) i.e., tn(n+3)-1 are given, the remaining}(n-l)(n-2) are also determined:

Or in other words,All curves of the nth deqreewhich pass through tn(n+3)-1

points pass also ihrouqh.Hn-l)(n-2) otherjixed. points.This theorem can be proved independently as follows :-Let < p = O and t / t = O be any two particular curves of thenth degree passing through tn(n+:3)-1 given points.

Then the general equation of a curve of the same degreethrough these points is < p + k t / t = O , where k is any arbitraryparameter. The equation is evidently satisfied by < p = O andt / t = O for all values of k, and consequently the curve passesthrough all the n2 intersections of < p and t / t , whatever be thevalue of k. Hence this curve passes not only throughtn(n+3)-1 given points, but also through the remainingn2-t?Z(n+3)+1 or -Hn-l)(n-2) points of intersectionof < p and t / t .

In fact, through the intersections of < p and t / t , an infinityof curves of the nth degree can be described, and anyparticular member will be determined by the condition that

This theorem also holds if the curve of the nth degree breaks npinto two carves of the kth and n-kth degrees respectively, when thecurve of the mth degree does not pass through all the intersections ofthe two curves-See Zeuthen-Su.r lu. determ.ination d'une cO!~rbealgehraique par des points donees. Math. Ann. Bd. 31 (1888).


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I......~ : : ' -r :GENERAL PROPERTllilS 29

it passes through any other point. But if this last-mentioned point is one of the common points of < p and t / t ,the value of k is indeterminate. It will be observed,however, that if n > 3, the above theorem is not always true.

EJ). 1. Straight lines are drawn through each of n collinear pointson an n-ie. Shew that these lines meet the 1/.ic again in points onan (n-I).ic.

Ex. 2. The taugents at four collinear points of a quartic meet thecurve again in eight points on a conic.

Ex. 3. All quartic curves drawn through 13 given points will passthrough three other fixed points.

Does this hold when the quartics are degenerate, each consisting offour right lines.

Note. This is the so-called Paradoc of Cm1JWr. Eulerhad already noticed that two curves of the nth degreeintersect in more points than are sufficient to determine anyof them. ' * Cramer t fully discussed this paradox but couldassign no reason for this. The first notion of systems ofcurves was given by Lame, and then Plucker in a note to"Entwicklungen" (Vol. I, p. 228) gave the above theorem.The algebraic aspect of the question was discussed in theworks of Jacobi, (Journal of Math., Vol. XV, 1841)and Plucker (Ibid, Vol. XVI, 1842).-8ee Plucker'sAlgebmischen Curuen, Einleitung; and also a Memoir byProf. Cayley-Cambridge Math. Journal, Vol III, p. 211.

26. If 1n>n, the theorem of 24 may also be statedin the following form :-All curies of the mili degree whichpass through mn-Hn-l)(n-2) points on a curve of ordern (m>n), pass also throuqli Hn-l)(n-2) Otk81' fixed. pointson this curve.

Consider a curve of the mth degree which consists ofthe given curve of the nth degree and another of degree

Enler-A Memoir in the Berlin Transactions for 1748-" On anapparent contradiction in the Theory of Curves.t Cramer-Introduction a I' Analyse des !ignes courbes algebriquae(1750).


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30 TlIEOltY oi! ' ptAN~ CURVESm-n. Let cp=O be the given curve of the nth degreeand 1=0 be a curve of order l1t-n, determined byHm-n)(m-n+3) assumed points. Now consider thecurve t { I f.cp=O. This curve passes throughn1n-tCn-I)(n-2) +H111,-n)(m-n +3) or tm(m+3)-1points and therefore, by the preceding theorem, it passesthrough Hm-I)(m-2) other fixed points. Of these latterpoints some will lie on cp=O and some on 1=0; and asmany will lie on cp as will make the total number ofpoints to nnt, and as many will lie on1as will make thetotal nurnber to lnem-n). 'rhus Hn-l)(n-2) of these}(m-l)(m-2) fixed points must lie on the curve cp=O.

E . 1 J . 1. Every quartic curve drawn through eleven fixed points on acubic passes through another fixed paint on the cubic.

This follows from the present theorem by putting In=4, ,,=3.We can prove this as follows :-

Every quartic curve through these eleven points aud two otherassumed points passes through three other fixed points. Now thegiven cubic and the line determined by the two assumed points makeup a quartic throngh these thirteen points. Therefore this systemmust pass through three other fixed points, of which only one can lieon the cubic, for otherwise a quartic through the points would meetthe cubic in more than 12 points.Ex. 2. A sextic 4>intersects a. curve of the eighth order > / I in 48

points. But in order to determine >/1=0, 44 points a.re required. Ofthese points if 38 are given, the remaining 10 are determined. A systemof curves > / I through these 38 points on 4>will pass through 10 otherfixed points on the same.

27. We have seen that if a curve of order m passesthrough mn-Hn-l)(n-2) points on a curve of order n,it passes also through t(n-l)(n-2) other fixed pointson the same. Hence it follows that, of the im(m+3) pointsreqnired to determine a curve of order m, if more thanmn-t(n-l)(n-2) lie on a curve of order n (m>n), thecurve cannot be a proper curve. For, suppose one moreof the points lies on the curve of the nth degree; in that


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GENERAl. PROPERTTER 31case the curve of the mt.h degree would meet the curve ofthe nth degree not only in nm-Hn-l)(n-2)+1 points,but also in t(n-I)(n-2) other points, thus making thetotal number to nm+l, which is impossible. If, however,the additional point be one of the Hn-I)(n-2) fixedpoints, the curve cannot be uniquely determined; for thenumber of independent points is nowmn-t(n-I)(n-2) + {tm(m+!3)-mn+ Hn-I)(n-2)-I},or, tm(m+3)-I.Hence we obtain the theorem-

Of the points determining a proper cw've of the mthdeg?'ee,the g1'eatest number which can lie on a louer CU1'veof order n is nm-Hn-l)(n-2),

28. CHASLES' THEOREM.""If a CU1'veof the third order pass th?'01tyheight of the

points of intersection of two curves of the third order, itpasses through the ninth point of intersection.

This in fact is a particular case of the more generaltheorem, given in 25, Chasles has given the followingproof ;-

Let U=O and V=O be any two curves of the thirdorder drawn through eight given points. The generalequation of a curve of the third order through these eightpoints is of the form U=~'V, when k is indeterminate.If this curve passes through a ninth point (x', y'), we musthave U'=kV', where U' and V' are the values of U and Vwhen ;t' and y' are substituted for . 1 ' and y in them. Then theequation of the cubicbecomesUV'- U'V=0. But this equationis satisfied by all the nine points of intersection of the twocurves U and V, and therefore this curve passes not only

* ChaBleB-Memoires de Brueellee, Vol. XI, p. 149.


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32 THEORY OF PLANE CURVESthrough the eight of these points of intersection, but alsothrough the ninth.

Chasles has used this theorem in demonstrating Pascal'stheorem 011 Hexagon, namely,-the three interseotions ofthe opposite sides of any hexagon insoribed in a conicsectionare in one right line.

Consider a hexagon inscribed ill a conic section. Theaggregate of three alternate sides may be regarded asforming a curve of the third order, and that of theremaining sides a second curve of the same order. Thesetwo intersect in nine points, namely, the six angular pointsof the hexagon and the three points which are theintersections of the pairs of opposite sides. Now the conicsection and the line joining two of these intersectionsconstitute a curve of the third order passing through eightof those nine points. Therefore this passes also through theninth point by the present theorem, i.e., the line passesthrough the remaining point, since the third intersection cannot lie on the conic.

29. The following important theorem* was given byM. Ger-gonne:-

If of the n" points of intersection of two curves of ordern, nm lie on a ourveof orderm (m


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GENERAL PROPERTIES 33

f passes through all the n9 intersections of U and V by 25.But f consists of an m-ic c p , which already passes throughmn of these n9 points, and cannot therefore pass throughany more. Consequently, the other curve if t of order (n-m)passes through the remaining n' -mn, or, n (n-m) points.

It will be observed that f passes through the remainingt(n-I)(n-2)-t(m-I)(m-2)

intersections of U and V. But these latter points cannotlie on c p , which already meets the system of the nth degreecurves in nm points. Hence they lie on ift=O, whichtherefore passes through-

t(n-m)(n-m+3)+t(n-I)(n-2)-t(m-I)(m-2)i.e., n (n-m) points.

Ex. 1. If of the nine intersections of two cubics, six lie on a conic,the remaining three are collinear. (Put n=3, 110=2.)Ex. 2. If of the sixteen intersections of two quartic curves, eight

lie on a conic, the remaining eight lie on another conic. (n=4, m=2.)Ex. 3. Any quartic through the intersections of a conic and

quartic meets the quartic again in eight points on a conic. (n=4, 110=2).Ex. 4. Any quartic through the intersections of "a cubic and

quartic meets the quartic again in four collinear points. (n=4,11O=3.)Ex. 5. If a polygon of 2n sides be iuscribed in a conic, the n(n-2)

points, where each odd side intersects the non-adjacent even sides,will lie on a curve of the (n-2)th degree.

This is a particular case of the present theorem when 110=2, andU and V are degenerate curves, each consisting of n right lines.

The product of all the odd sides may be regarded as one system ofthe nth degree, and the product of the even sides another. These twosystems intersect in n' points, namely, the 2n vertices of the polygonand the remaining n(n-2) points which are the intersections of the nodd sides with the (n-2) non-adjacent even sides of each, since eachodd side has two adjacent even sides. Now, the 2n vertices lie on aconic, and therefore by the present theorem, the remaining n(n-2)points lie on a curve of the (n-2)th degree.


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34 THF.ORY OF PT .'\.NF. CURVES:30. Prof. Cayley has given the following generaltheorem, ' * ' of which the theorem of 26 is a particular

case:-If a cl1,rveof the rih. order (r not less than 111 01' 1L, notgreater than (m+n-:3) pass throuqh.

mn-Hm+n.-?'-1)(?n+n-r-2)of the points of inte?'sectionof 1'11'0 curves of the mih. and nthorders respectively, it passes throuqh. the remaining

t(m+n-T-l)(m+n-?'-2) points of intersection.Let U=0 and V=0 be any two curves of orders m and n

respectively.Select a group of -Kr-m)(r-m+3) arbitrary points onV and describe a curve .p=0 of order (r-m) through them.Again, select a group of H1'-n)(r-n+3) points on U

and describe a curve 1/1=(1 of order r-1 through them.Now consider.the two curves of order 1', namely,

The curve f passes through H?'-m)(r-m+3) pointson V, t(1'-n)(1'-n+3) points on U, and all the inter-sections of U and V, i.e., through

{t(r-m)(?'-m+3) +n~n+H'r-n)(r-n+3)}or {t?{r+3)-1}+-Km+n-r-l)(m+n-?'-2) points.

Again, the curve f' passes through the same

Hence. if we select a group ofmn-t(m+n-r-l) (m+n-?'-2)

points of intersection of U and V, then the curves fto Cayley-Collected Papers, Vol. J, No, 5, p, 25.


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~\,'..

UENERAL PROPERTIES 35and f' will pass through ir('y+3)-1 points. Therefore,every curve of the rth degree which passes throughthese ~r(r+3)-1 points will pass through all the intersee-tions of f and f, and consequently through all the 11mintersections of U and V. But

mn-t(m+n-r-l )(m+n-r-2)of these 'Innintersections lie on the curve. Therefore itpasses through the remaining H?n+n-r-I)(1n+n-T-2)intersections.Note. It will be observed that r cannot be less

than m or 11, it can at the most be equal to the greaterof 1IL and n. In order that the curve c f > may be distinctfrom, and does not include as part of itself, the curve V,'r-m must be less than n, i.e., r must be less than m+n.If 'y=m+n-l or 1n+n-2, the theorem is meaningless.Hence r must not be greater than m+n-a. See Bacharach,Math. Annalen, Vol. XXVI (1886), pp.275.299.

If m=n=r, we obtain the theorem of 25.Ea;. 1. All quartics which pass through eleven points of intersection

of a cubic and a quartic also pass through the remaining intersection.[Put r=m=4 and 1I=3J.

Ex. 2. If ofthe twelve intersections of a cubic and a quartic, six lieon a. conic, the other six will also lie on another conic and the fourpoints in which the two conics meet the quartic again are collinear.

Consider the quartic consisting of the given conic and anotherdetermined by five of the remaining six points of intersection. Thisquartic therefore passes through eleven of the points of intersection ofthe given cnbic and the qnartic. Therefore it passes through theremaining intersection, whioh must lie on the second conic, sincethe given conic meets the cubic already in six points. Hence theremaining six points lie on a conic.

Again, suppose that the first conic intersects the qnartic in thetwo points A andB, and let the line AB meet the quartic in twoother points C and D. Now consider the quartic consisting of thecubic and the line AB. This intersects the given quartic in 16 points.


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36 TH~OnY OF PLANE CURVESBut the quartic consisting of the two conics }asses through fourteenof these points. Therefore it must pass through the remainingtwo points C and D. Consequontly C and D lie on the second conic.Hence the conics intersect the quartic in fonr oollinear points.

This follows from the presen ttheorem, as a particular case, byputting m=4, n=3, and r=4.Ex. 3. A curve of-the fifth degree which passes through fifteen ofthe intersections of two quartics passes also through the remaining

intersection.Thi~ follows from the present theorem by putting r=5,m=n=4.Ex. 4. The tangents at the six intersections of a conic and a cubic

meet the cubic again in six points on a conic.

Ex. 5. Show that a quartic drawn through the intersections oftwo cubics meets either cubic again in three collinear points.Examples 3 and 4 of 29 can be deduced from the present theorem.

Sec. II: Theory of Residuation.31. We have seen, 26, that if of the 'Inn intersections

of two curves of the mth and nth degrees (m>n),mn-t(n-l)(n-2)

are given, the remaining Hn-I)(n-2) are determined.Hence we see that the two groups of points are notindependent, but together form the complete intersectionof two curves of orders m and n respectively. Theseconsiderations led to the development of a theory of groupsof points on a curve, the principles of which are containedin a paper by Noether.* The principles of the theoryare illustrated in the following theorem :-

If a group of n(l+m) points on a curve C. of ordern forms the complete intersection of C. with a curveC I +,. of order (l+m), and nl of these points form thecomplete intersection of C. with a curve C I of order l,

< Noether-" Ueber einen Satz ans der Theorie del' AlgebraischenFunctionen "-Math. Ann. Bd. 6, 1872, and" Zum Beweise des Satzes,etc.," Math. Ann. Bd. 40, 1892, p. 140.


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1 " ;

RESIDUATlON 3 7

then the remaining nm points form the complete intersectionof C. with a curve Cm of order rn.

The theorem of 29, given by M. Gergonne, is only aparticular form of this theorem, when l+m=n.

Professor Sylvester * has propounded this remarkabletheory, called the theory of residuation. We give belowcertain definitions:

(1) The curve C. is called the basis-curve.(2) Anyone of the groups mn and nl points is called a

residual of the other, and the two groups are saidto be residual. to each other.

(3) Any two groups of points on a basis-curve whichhave the same residual group are said to becoresiduai to each other.

(4) If a group is the complete intersection of a curvewith the basis curve, it is said to have a zeroresidual.

32. In the light of these dein itions the above theoremmay be stated in the following for m :-If a gl'OttP of n(l+m) points on a Cltrve of order n has a

zero residual and nl of these points have also a zero residual,then the remaining nm points have a zero residual,

Consider the curve C!+m =C"+IO_nC. + C',.C,=O,l+rn>n, which is of order (l+m) and passes through theintersections of C 1 and Cn' This equation is satisfied forall points where C. =0 and C'mC1 =0. Now the curvesC,+ and C. intersect in n(l+l1t) points, of which In lie onthe curve C 1 Therefore the remaining nm points lie on thecurve C'of order m.

If l+m=n, then we have simply to put-C', +_. =C' 0=a constant.


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: 3 8 TlltORY O}' PUN}; CUltvESThe proof can be equally applied to the case when

Z+m


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rRESIDUATION ~9

points of intersection of C", ana 0.. Hence the curveC! + ' " passes through all the points of the two groups[L] and [M], and since the number of points in the twogroups is n(l+m), the points of the group [L+MJ arethe complete intersection of C 1 + ' " with C .

Therefore we have [L+MJ=O.Case II. If l+m=n, l+m-n=O and 0',+ _. is to beput equal to a constant, and the same argument holds.

Oase III. If l+m


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40 THEORY OF PLANE CURVES

the remaining mn points must lie on C' which is of degreem, and consequently the mn points [M] are the completeintersections of C'm with CnJ and :. [M] =0.

35. MULTIPLICATION 'l'HEOREM:If [L]=O, then [kL]=O, where k is a positive integer.Let C" be the basis curve, and let C /, a curve of order

l, cut C" in the zero residual group of points [L], namely,[L,. L" ...L"d=O.

Consider another curve C 'l contiguous to C / . Thiscuts C" in a group of points

which is also a zero residual.Hence, by the addition theorem, we have

or [L, +L'" L. +L' , ...L" I +L'. d =0.But since C I and C'/ are contiguous curves, the points

are contiguous in pairs, sothat ultimately L 1 and L'" etc.,are coincident. We therefore obtain-

or, [2L]=0.Repeating the same process for another series of conti-

guous points, we obtain [3L] =0.Continuing in this way k times, we finally obtain

[kI .=O, where k is any positive integer.Note. It is to be understood in this theorem that when

a point is multiplied by a positive integer. consecutivepoints on the curve are taken into account.There is no division theorem in the thelYl'Yof Residttation.


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RESIUUATION 41Ec, 1. If of the nine intersections of two cubics, six lie on a eon ic,

the remaining three lie on a right line.Let the six points which lie on the conic be denoted by [LJ and

the remaining three by (M].Then [L+1Il]=0, and also [LJ=O :. [MJ=O.

i.e., the three points [lI!J are the complete intersections of thebasis cubic with another which must evidently be a right line.Ex. 2. If eight of the sixteen intersections of two quartics lie

. on a conic, the remaining eight lie on another conic.Let the eight points on the conic be denoted by (LJ and

the remaining eight by [M]. Theu [L+MJ=o. and also [LJ=o.,:. [iIlJ =0..

i.e., the eight points [MJ are the complete intersections of thebasis quartic with a curve of degree nt, such that 4m=8, and con-seqnently m=2, i.e., the eight points lie on a conic.

36. If we have the two residual equations [L+ NJ=0and [M+NJ=O, when L,M,N are groups of points ona basis curve, then we obtain by the subtraction theoremthe formula [L-MJ=O, or, [L]=[M], wMch signifies thatthe two groups [L] and [M] are coresidual with thecommon residual group [N]. If further [L']=[M'J withN' as a common residual, then we have

for, we have[L+L']=[M+M'J;

[L+K]=O and [L'+N']=O:. [L+L'+N+N']=O.[.M+M'+N+N']=O.[L+VJ and [M+1.l']

Similarly,Henceare coresidual with the common residual gl'OUp [N+N'J.Further, we have

[L+:M']=[L'+M], or, [M-M']=[L-UJ.Again, the equation [L+M-r"J=[NJ signifies the

same fact as [L+MJ=[L+N], which is the same thing


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l42 THEORY OF PLANE CURVES

From all these it is seen that we can apply the rulesof addition and subtraction to the residual equations. Wegive below two important theorems of frequent applicationin the theory of residuals:

(1) If two groups [L] and [M] be coresidual, anysystem [N ] which is a residual of one will be a residual ofthe other.

For we have [M-L]=O and [L+NJ=O.:. [M+N]=O.

(2) Two groups which are coresidual to the same groupare coresidual to each other.

Let the groups [L] and [N ] have a common residual[L'], and let [M] and [N] have a common residual group[M/J, so that [L] and [M] are coresidual to the same group[N]. Then the groups [L] and [M] are coresidual to eachother.We have (1) [L+L'J=O,

(3) [M+M/J =0,(2) [N+L'J=0,(,j,) [N+M'J=O.

From (2) and (4) we have[L'-M/] =0,

and therefore from (3),(5) [L'+M]=O,

i.e. [L'J is residual to [M1 .Similarly, (6) [M'+L]=O

i.e. [M'] is residual to [LJ.Now, from (1) and (5), [L] and [MJ are coresidual

with the common residual [L']. Similarly from (3) and(6), [L] and [M] are coresidual with the common residual[M'l

J


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1l.ESIDUATION 43Oor. If two coresidual groups consist each of a single

point, the two points coincide.Ex. 1. If of the 3 (m+ n) intersections of a curve of the (m+ n)th

order with a cubic, 3n lie on a curve of the nth order, the remaining3mlie on a curve of the mth order.

Let the two groups of 310 and 3mpoints be denoted by [LJ and [MJ,respectively.Then [L +MJ=O and [LJ=O :. [MJ=O. i.e. the 3n. points in [MJ

are the complete intersections of It cubic with a curve of order k, suchthat 3k=3m. :. k=m, or the curve is of order m.

Ex. 2. The tangents at 1b collinear points of an n.icmeet thecurve again inn(n-2) points lying on an (n-2).ic.

Ex. 3. If six of the intersections of a oubioand a quartic lie on aoonio,the other six lie on another conic,

37. BRILL.N5ETHER'S RESIDUAL THEOREM: ' * 'If two groups of points [L] and [L'] are coresidual with

a common residual [M], and if [M'] is any other residualof [L], then [M'] is also residual to [V]; [L] and [L'] areconsequently coresidual with respect to [M'], and to eachresidual of any of the two groups [L] and [L'J.

Consider the four groups of points [L], [L'], [M],[M'] on a basis curve. Now, if

[L+M]=O, [L'+M]=O, and [L+M']=O,then we have from the first and third [M]=[M']. Substitu-ting this in the second we obtain [1/ +M']=O.

i.e, [M'] is a residual of [L']... [L] and [L'] are coresidual with the common residual

group [M'],This theorem has been generalised as follows :-If the groups [L,}, [L.], [Ls],...[L,'] on a basis curve

are ooresidual with a common residual group [M], thenthey are also coresidual with respect to any other residualgroup of anyone of them.


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4~ THEO~Y OF PLA~E CURVES38. EXTB~SION OF THB RESIDUAL THEORE)! :The direct proof of Brill-Noether's theorem depends

entirely on a similar identity as used in the proof of 32,which is evidently a particular case L C-,l,-~--.Mof the theorem. We can, however,represent this residual theorem bymeans of the adjoining diagram. BASIS CURVE CIn,e n

By introducing a new auxiliarycurve O'-n', we may express theidentity in the form:

M'L---::,-----'L.'c ; ,(1)

The orders of these curves must then satisfy theconditions

m+l1~'=n+n'=l+l' (2)Note. If the identity (1) holds, the theorem is proved.

'I'his is the case in particular, if the curves O m , O n' , 01 , 0 '1 'are" adjoints " to O f t , i.e., if they pass (i-I) times throughan i-pIe point of Cf t . In the theory of groups of points ona curve, " Adjoint" curves play an important part. \Ve,however, postpone discussion till a later chapter.

In the identity (1), the curve 0' n' may as well be takenas the basis curve, and it may be treated exactly as thecurve O f t .

Since Om, 01,t o ,

BASIS CURVE err{e " ,M ' , . .- - . , . . . . . . - - ~ ,C : ,

Om ' , and O i' intersect mutually ;in theother four groups of points Ll' MI'L'1> 11'['" which by (1) lie on O 'n'they may be represented by meansof the accompanying diagram.

Therefore, we may write Noether'sidentity in the followiug symmetricalform:

Cm0",'+ o t o- + O n Cn'::;:O_ (3)


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RESlDUATION 45'rhus, supposing we deal with singular curves, we recog-

nise the completely identical nature of all the six curves onwhich lie the foul' groups of points satisfying Brill-N oether'stheorem. From this extension a number of other theoremson configurations of these points can be deduced."

We can represent theidentity by means of a cubeas well, shown in the dia,gram, if the squares in thepreceding diagrams aretaken as its base and top'The eight groups of pointswill be at the corners, andthe six curves will be takenas corresponding to the sixfaces.

M'

If we take LM, LM' and LL1 as the axes of a " y, and z,and the cube of unit edge, the faces X = O , ;c=l, y = O , y=l,z::::O, z=l will represent the curves Gm, Cm', O z , C l',Cn, C n' respectively. Each curve passes through thepoints lying at the four corners of the correspondingface. The complete intersection of any two curves consistsof the points at the ends of the common edge of theadjacent faces representing the curves. Thus, the curve(J I passes through the pointsL, M, Ll M1; and thecurves C I and Cm intersect in the points L, L"and so on.

39. We may illustrate the above facts very clearly bymeans of the diagram given on the next page.

Each curve is replaced by a circle and each group ofpoints is denoted by a sing'le corresponding point.

Instead of starting with one curve as a basis curve, wemay start with one group of points, and state the theorem inthe following form:

* E. Study in Marburg-e-" Ueber Sohnittpunktfiguren ebener algeb


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46

r

THEORY O:r PLANE CURvES

If three algebraic curves Cm , C/, C. pass thl'ough thesame group of points L and intersect by pairs III thegroups IJI' M ' and M, thenthree other curves Cm',Cr , On' can be determinedin various ways, such thattheir orders satisfy equa-tion (2) only, and theypass through the groupsM, M / , r., respectivelyhaving the same residualsM'l> M, and L' on C m , c.,C. in pairs, and whoseremaining points of inter-section together form a single group L'1'

Ex.!. Three conics through the same two points I and J intersectin pairs in three other pairs of points. The three lines joining themare concurrent.

Ex. 2. A conic meets a quartic in eight points. If another coniotouches the quartic at four of these points, a third oonio touches thequartic at the remaining four points.

E. 3. A conic passes through four fixed points of a cubio. Showthat the line joining the other two intersections of the oonic and cubicmeets the oubic again at a fixed point.

E :. 4. Through a group of points P on an n-ic a curve is drawnto intersect the n.ic ag-ain in the group P" through P, is drawn ~,curve to intersect the n.ic again in the group P z, and so on. Showthat if the final group consists of a single point, the same point will beobtained whatever be the curves used.

Ex. 5. If three cubics pass through seven common points, the linesjoining the remaining intersections of the cubics taken in pairs forma triangle whose vertioes lie one on each of the three given cubics,

Ganguli, Theory of Plane Curves, Chapter 2

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