Ganguli, Theory of Plane Curves, Chapter 12

8/14/2019 Ganguli, Theory of Plane Curves, Chapter 12

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CHAPTER XIITHEORY OF CORRESPONDENCE.

246. CORRESPONDENCE OF POINTS ON A CURVE:In Chapter X, we have discussed the general principles

of correspondence of points of two different planes, or ofthe same plane, as a whole. In this Chapter, we shalldiscuss the correspondence of points on the same curve,or on different curves.

The simplest of such correspondences is illustrated bythe homographic systems of lines and conics, the essentialsof which are to be found in all treatises on conic sections."In fact, there is a (1, 1) correspondence between the elementsof two bases defined by the bilinear relation

AU'+BA+CA'+D=Owhere A and A' are the parameters of the correspoudingelements.

When the two bases are superimposed, we obtain acorrespondence between points of the same base, specialcases of which are studied under the name Involutionranges or pencils.

Chasles extended and discussed the theory as appliedto unicursal curves; but the theory is applicable toall curves generally, and we shall presently consider thegeneral principles of correspondence of points on the samecurve.

Let P and P' be two points on the same curve, such tha.tto any position of P there correspond r' positions of P', andto a given position of P', r positions of P. Then the pointsof the curve are said to have an (1', 1") correspondence.If 1'=1"=1, the correspondence is .(1, 1), and rational.

Scott-Modern An. Geo., 192196.


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808 THEORY OF pLANE CURvES

The correspondence ' * ' between points of a curve may beinstituted in various ways, as the following illustrationswill show:

(1) The points of a conic may be put into a (1, 1) corres-pondence, if the points collinear with a given point in itsplane is made to correspond, the points forming an involu-tion range on the conic.

(2) Any radius vector through a fixed OrIgm 0meets a curve of order n in n points P, Pl, P" p.,...p.-l IfP" denotes anyone of the points P l' P ., ...P0-1 we may saythat to a given position of P there correspond (n-I) posi-tions of P', and conversely, to any position of P' there are(n-I) positions of P. Hence there is an (n-I, n-I)correspondence.

This will be a rational correspondence, if n=2 (Case 1),a particular case of which is afforded by the circularinversion 15 & 217.

The principles of correspondence for points in a line wasestablished by Chasles in his paper in the Comptes rendus, June-July1864. But prior to him De Jonquieres considered the principle in1860-" L'reuvre mathematique d'Ernest de Jonquieres." It has beenextended to unioursal curves by Chasles in a paper of his-Sur les courbesplanes 01 a double courbure dont les points peuoen: se dete>-miner indivi.duallement-Application du principl~ de c01Tespondence dans la theoried~ CBS courbes"-Comptes rendus, Vol. 62 (1866), p. 534. Cayley referredto the principle-Comptes rendus, Vol. 62 (1866), but gave a discussionin his memoir-On the correspondence of two points on a curve (Coli. Works,Vol. 6, p. 9), where he discussed only II. particular case. Finally he gavea number of applications in his paper-Second Memoir on the Ourueswhich satisfy gillen conditions-Coli. Works, Vol. 6, pp. 263271. But analgebraio demonstration of a more general formula has been given byBrill-" Ueber Entsprechen von Punktsystemen auf eine,' Ourve "-Math.Ann., Bd, 6 (1873), p. 33, and a geometrical treatmeut in his paper-"Ueber die Correspondenzformel "-Math. Ann., Bd. 7 (1874), p. 607.Lindemann has given a demonstration of the principle with the helpof Abelian integrals-Crelle, Bd. 84 (1878), p. 301. See also a paperby G. Loria-Bib). Math. (3) 3 (1902), p. 285.


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THEORY OF CORRESPONDENCE 309When the origin 0 lies on the curve, there will be an

(It-2, n-2) correspondence; and in general, if 0 is anr-ple point on the curve, there is an (n-r-l, n-r-l)corresp ondence, i.e., corresponding to any position of P thereare (n-r-l) positione of pi, and viceversa.

It is to be noted that a (J, 1) correspondence is possiblefor a cubic or a quartic with a node, but it is not alwayspossible.

247. In the preceding illustrations, the correspondenceIS symmetrical, i.e., from either given point the other isobtained by the same construction. But there are corres-pondences which are not so symmetrical. The followingillustrations will clearly explain what we mean;

(1) There may be instituted a (1, 2) correspondencebetween the points of a conic as follows ;-

Let 0 be a point on the conic Sand p a line in the sameplane. A radius vector through 0 will meet the conic ina point P and the line in a point. Q. The polar line of Qwill meet S in two points P1 and P,. Thus there IS an(1, 2) correspondence between P and P 1> r - , (PI).

(2) The tangent at any point P of an n-ic meets thecurve in (n-2) other points, so that to any positionof P there correspond n-2 positions of Pl. But if pi isgiven, P may be anyone of the points of contact of the(m-2) tangents which can be drawn from pi to the curve,m being the class. Thus to any given position of pi thereare m-2 positions of P, and there is consequently an(m-2, n-2) correspondence.

248. ANALYTICAL DISCUSSION:The preceding examples show that a geometrical construc-

tion can be given for determining a correspondence on a givencurvef=O.

An algebraic correspondence (l, Ie ) between the pointsP, pi of two curves, or of the same curve, is defined by a


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'

310 THEORY OF PLANE CURVESsystem of algebraic equations between the coordinates ofP and pi, such that when P is given, there are k points pi,and when pi is given, there are l points P. In fact thecorresponding points are obtained as the intersections ofa certain curve @ with 1=0. In the above illustrationsthis curve @ was taken to be a right line. It may againhappen that some of the intersections pi coincide with' thegiven point P, and these points must then be excluded.

Let the equation of the curve @ be given in the formIii' [(, Y ")l . (t1 y ' ,1)kJ-OJ" ," , " ,IJ _

which contains (.e, y , e ) in degree land (.t/, y ', Zl) in degree k,If p (e, y , z) be given, this equation represents a

curve @k of order kin C u ' , y ' z') ; and if P'(z', y ', z') be given,a curve @ I of order I in (J', y, z) . The two curves @land @k intersect the given curve 1=0 in In and lcn pointsrespectively, and therefore weobtain a (In, kn) correspondence.If, however, the curve @l for a given point (e', y ', Zl) passesthrough the same, that point is to be excluded, so that if aintersections of @l with 1coincide at P ' ( X ' , y ', e' ) , the numberof remaining intersections P is In-a, Similarly, if thecurve @k for any given point (a; , y, z) meets 1=0 in f3points coinciding with (e, y, z), there are kn - f3 remainingintersections pi, Thus we obtain a (In-a, kn-f3) corres-pondence, and we denote it by (In-a, kn-f3).

Ere . In EiD. 2 246, we havecorrespondence is (n-I,n-I).

In Ere . 2 247 I n=m, k=l, =,8=2, and the correspondenoe is(m-2, n-2).

l=k=l , also =,8=1, and the

249. UNITED POINTS:As in the case of general correspondence, we have

elements coinciding with its correspondents, so in thisparticular case, a point may correspond to itself, and isthen called a ?tnited point. For example, in the case of


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THEORY OF CORRESPONDENCE 3 1 1

involution range of points on a conic ( E : ! . 1 246), the pointsof contact of the tangents drawn from the fixed point 0to the conic are united points. In general, if the corres-ponding points are collinear with a fixed point, the unitedpoints are the points of contact of the tangents drawn fromthe fixed point to the curve. Bence the number of suchpoints is equal to the class of the curve.

Prof. Cayley "" explained by means of a number ofillustrations the general formula for finding the unitedpoints without any formal proof. But later on Brill t gavea formal accurate proof of the formula.

In the equation of 248, if we put ;c=.v', y=y', z=z',we obtain an equation 1+k=0 of order l+k, which nowrepresents a curve of order l+ k passing through all thosepoints for which k passes through ( :1 ', y , z) , and l passesthrough (m ', y', z') . Thus the curve 1+l=0 intersects thecurve 1=0 at the united. points. Therefore, the correspon-dence (In, kn) r= (a, b) has (l+k)n=ln+knr=a+b unitedpoints as given by Chasles.

If, however, k meets 1in one or more points (a) coin-cident at (x, y , z) , and I meets 1in one or more points ({3)coincident at (x', y', z'), which, of course, happens, whenthey are singular points on I , these a and {3points arenot to be included in the number of united points, and theorder of multiplicity in a and {3diminishes the number ofsuch points.

In the preceding examples, a=f3=l, and the formulaholds for rational curves. In fact, when a={3, the investi-ga.tion becomes much simplified and the correspondence

* Oayley-" Note sur la correspondence, etc." Oomp.Rend. Ac. So.,Paris (1866), Vol. 62, pp. 586590,also" Correspondence of two pointson a ourve "-Call. Works, Vol. 6, pp. 913.t Brill-Ueber Entspreobeu van Punktsystemen auf einer Ourve-

Math. Ann. Bd. 6 (1873), pp. 33.65, aud " Ueber die correspondenz.formel "-Mnth. Ann. Bd. 7 (1874), pp. 607622.


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312 THEORY OF PLANE CURVES

depends upon the value of a, and is then denoted by thesymbol (In-a, kn-a) c.'

In the example (1) of 247, there are three unitedpoints, namely, the points M Il M O l where p cuts the conicand the point O.

Again, P1may coincide with 0, while Psis distinct, orboth P" P, may coincide with M" M.; but this does notstipulate any higher species of united points, as has beenshown by Cayley in the general case.*

250. CHASLES' CORRESPONDENCE THEOREM:Let Cn and Cn' be two curves of orders nand n'respectively, such that there is a (1, 1) correspondence

between them, i,e., to each point P of CK there correspondsone and only one point pi of Cn', and vice versa. Let usfirst determine the class of the envelope of P'P', or in otherwords, let us see how many of the lines, such as P'P', passthrough any point (0, 0, 1) for example.

Consider a line p through the vertex 0 (0,0, 1) whichintersects CK in n points, corresponding to which there aren points on Cn'. The n lines joining 0 to these n pointson Cn' are given by an equation of the form-

Any other line q through 0 will similarly correspond to nlinea determined by the equation-

The pencil of lines P+Aq=O, which, for simplicity, maybe taken identical with X+AY=O, then determines aninvolution pencil of order n,

(1)

Ca.yley-Seoond] memoir on curves ;whioh satisfy given eondi-tions-the principles ofoorrespondence-Coll. Works,Vol.6(1868),p. 265.


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THEORY m' CORRESPOXlH~NCE 313Similarly, considering- the intersections of the lines 'P

and q with en', we obtain an involution pencil of order n',(2)

where t f r n ' and t f r ' " , are bina-ry expressions of order u', similarto 1 > " and 1 > ' " .

'I'ho two involutions (1) and (2) are then projective andhave the same vertex. The double or self-correspondingelements are obtained by eliminating ,\ between (1) and (2)in the form

(3)The equation (8) is of order n+n', and therefore gives n+n'self-corresponding' rays of the two pencils, whence theclass of the envelope of Pf" is determined.

The above considerations hold for two ranges of pointsas well, and we obtain Chasles' Correspondence Theorem:

If there are two superimposed systems of elements, suchthat to each element of the first aystern correspond n ele-ments of the second, and to each element of the second, n'elements of the first, then there are n+n' self-correspondingelements; or in other words:

In an (n, n') correspondence on the same base, there aren+n' double elements, as we have otherwise determinedin the preceding article.

251. CORRESPONDEXCE IXDEX OR CHAHACTERISTIC :Let A and A' be any two points in the plane of the

n-ic /=0, and draw a line p joining A to any point P(e, y, z) on /=0. Determine the points pi (,,/, y', z')on / which correspond to P, and to all other points where theline P meets the n-ic, and join these points to A' by meansof lines p'. 'I'hen the locus of Intersection of the corres-ponding lines P and p' is a curve , which can as well beobtained if we start with a point pi (.t', y', Z ' ) . 'I'his curve then meets the curve f=O at thennited points. If a unitedpoints coincide at each point P(x, y , z) , must paRS

40


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314 THIWll Y OF PLANE CUlt VES

through these points, and must contain f as part a times.'fher-efore, to each line p ' the line I' corresponds a times,and these a-pie line P passes through P' (x', y', z'). Hencewe have a=(3 ( 248).

But since A and A' may be any points in the plane, werequire only to determine how many of the points (,,,',y', z'),common to C P k andj, coincide at (,I', y , z). The number aof these points is called the" inde " of the correspondencewhich is denoted by-

(In-a, kn-a)", or, (a-a, b-a)".252. COMMO}! EL~;~[ENTS OF 'I'wo CORRESPONDE}!CES :Let us consider on the curve j=O the two correspondences

c p = ( a , & ) 0 and c p ' = ( a ' , b')oin both of which the correspondence index or characteristicIS zero. We shall now determine the pairs ofpoints (x , y, z) ,(a:',y', z') on j, which simultaneously satisfy the two corres-pondences, i,e., we shall find the number of points of thetwo correspondences which correspond to the same point(xu Y " : , ) on f. This number is determined by the intersec-tions of j with a certain curve I { r .

To determine the order of the curve I { r , we count thenumber of points in which any line L will intersect it.In virtue of the second correspondence c p ' , to any point(x, y, z ) of this line there correspond b ' points (xu Y " z,)on j, which are obtained by its intersection with a curve /.To each of these b' points correspond, in virtue of the firstcorrespondence c p , b'l points ( . v ' , y', z ' ) on Ij given by the curve I' To each of these points again correspond in the same wa,yal k points ( . , y, z ) on 1 . 1 . Consequently, on the lineL we havea (b'l, a'k) correspondence of points (x, y, z) and (x', y', z').The order of the curve I { r , which may be considered &iigenerated by two pencils of lines, is consequently (b'l+a'k).

Hence, I { r intersects j=O III n(b'l+a'k)=ab'+a'hcommon points (a: , y , z) and (a::', y ', z'), or, ab'+a'b pairs of


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THEORY or CORRESPONDENCE 3]5points (,1', y, z), (e', y', Z l ) , which satisfy both thecorrespondences. 'I'he same result may be obtained in adifferent manner as follows:

To each point ( . 1 ; ' , y', Z l ) of the 'plane correspond thell' points of intersection (. t ' , y , z) of the curves 1 > 1 =0 and1 > ' , ' =0 belonging to the two correspondences. Similarly, toeach point (,[',y, z) there are kk' points (,e ', y', Z l ) . If (.t', y', Z l )moves along a line, the ll' correspondents ( , 1 ' , y, z) move alonga curve of order (k'l+kl'). Consequently, if (x', y', zl)des-cribes the curve j, (.r, y , a ) describes a curve of ordern(k'l+kl').

Each point of intersection (;, y, z) of this curve with jtogether with a point (;I', y', Z l ) on j will give the requiredpair of points. The number of such pairs satisfying boththe correspondences (a, b ) , (ai, b ') on j is, therefore, equal to

n' (k'l+ 7 . : 1 ' ) =(nk.lll' +n7


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: .H6 TH~ORY OF PLAN~ CURVESThis is a quadratic in "and therefore gives two values" 1 and ". of"for which fL=fL'. If" is eliminated between e and e', we obtain abilinear relation between fL and fL', of the form:

1" I ' - + z , ('fL+U 1 = 0a'f l.1 + b' c'p.' + 0' I

Hence, fL, fL' determine a (1,1) correspondence. If we put fL=fL', W(1obtain a quadratic equation giving the united points of this corres-pondence. It eas ily follows then that the common pairs are given 1 ,)"

A l / l - l -u -, ,258. If one or both the correspondences have a point

where one or more corresponding' points coincide, then todetermine the common points ( . 1 ' , y, r), (x', y', z') of thecorrespondences, the number ab ' +a'b must be reduced.

Consider the correspondences-1>=(,b)o and 1>'=(a'-y', b'-y')-y'

In this case the formula of the preceding' article holds,if only distinct pairs of points are taken into account.Hence the number must be reduced by the number of coinci-dent corresponding points, but such coincidence takes placey'-times only at the united points of 1 > , at each of which < P 'has always a y'-point, and consequently it is equivalent toy'(a+b).

The number of distinct pairs of points which satisfysimultaneously the two correspondences is then equal to

ab'+a'b-y'(a+b) =a(b'-y') +b(a'-y')254. We shall now consider the two correspondences

of indices "I and y' respectively, i,e.,

1>=(a-y, b-y)~, 1>'=.(L'-y',b'-y'h'Let UH deform the correspondenee 1 > ' a little mto a newcorr-espondence 1 > , =(a', [,')0' 1';0 that the present case isreduced to the preceding one. 'I'he correspondence 1 > , has 110y'-point at (x)=(.,,') but g-ives, on the other hand, OlJ f


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THEORY OF CURRESPONDENCE 317

aaerres of y' points contiguous to all points a,' or x. 'I'henumber of pairs of points common to the two correspondences< P I and < P is, by the preceding article, equal to

ab'+a'b-y( a'+b')Among these pairs, there are t.hose consisting of two stillmore contiguous points ,1', :u', which will coincide, and indi-cate a further reduction, when < P I is again deformed to < p ' .

Let the point (''!) move on j, then they' points (.) of< P I become contiguous to it. But if (x') moves up to coinci-dence with a similar united point ( , 1 ' 1) of < P , then also theseries of points on < P I moves up to coincidence with ( . 1 ' 1)' andfinally coincides with it, and then moves a,my, as (,e')proceeds further.

Hence we conclude that the common pairs of pointsof < P and < P ' will be obtained by deforming < P I back to< p ' , each of the N united points of < P will occur 'I' times.

Therefore the number N of all the distinct common pairsof < P nd < P ' is given by-N=ab'+a'b-y(a'+b')-y'K

Similarly, by deforming < P to < P " we obtain-s-N=a'b+a!J'-y'(a+b) -'IN'

where K' denotes the number of united points of the cor-res-pondence < p ' .

By identifying these two expressions, we obtainN -(a-y)-(b-y) =N!~(c{-y')--.(b'-y') = = M (say)

'I Iwhich is independent of the intermediate correspondence.Therefore .Mdepends on I, and ill fact, on its deficiency andis to be determined by considering a special case.255. CAYLEY-HHlLf;~ COU!{ESI'O);[)BXC!': FORJIULA:

Let us take the correspondence between the point ofcontact (.,,') of a tangent to the curve 1=0 and the (n-2)other points (.r) where the tangent meets f. In this case


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318 1'HEURY OF PC,ANE CURVES

the number of united points is evidently equal tv the nurnbaj-of points of inflexion of j, i.e., :3n(n-2)and azzn; b=n(n-l), y=:!., ~'f :3n(n-2)-(n-2)-{n(n-l)-2} -( -1)(, -')-2l' 2 -----, - It /I ~- pwhere p is the deficiency.

Hence we obtain the correspondence Ior mula for unitedpoints of .p : N=(a-y)+(b-y)+2yp.*

Thus, for the correspondence . p = (a, (3)J. the number ofunited points on a curve j of deficiency p (ussumiug thereare only double points, etc.) is given hy-

N=a+(3+2yp tThis is known as Cayley-Brill's CorrespondenceFormula,

and it is easily seen that Chasles' Formula holds when eithery or p or both are zero.

If we substitute this value of N in one of the formulaefor N, we obtain for the common united pairs N ofthe two correspondences-

.k ( b ) .k' (' , b ' , )'1'= a-y, =n-, 'I'= a -y, -y'''y'N=ab'+a'b-y(a' +b')-y'N=ab'+a'b-y(a' +b') -y' {(a-y) + (b-y) +2yp}=(a-y )(b'-y') + (b-y)( a'-y') -2pyy'=a(3'+a'(3-2yy'p.

41 < For a complete discussion of the united points, etc., the studentis referred to Olcbsch-e-Lecons sur Ia Geometrie, Vol. II, Chap. I,pp, 146188.

t 'I'his formula was first given by Cayley-Compo Rend., Vol. 62(1866), p. 586, and Proc. Lond. Math. Soc., Vol. 1 (1866), p. 1, and waslater Oll proved by Brill-Math. Ann. Bd. 6(1873), p. 33, Bd. 7(1874),p. 607- Several other proofs, etc., were given by Bobek, Segre,Lindemann, Zeuthen, etc. See also Severi-Torino Mem., Vol. 50 (2)(1901), p. 82, Vol. 54(2) (1903), p. 1, and Torino Atti, Vol. 38 (1903),p.158.


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THEOR Y OF COHItJ


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320 THEORY OF PLANE CURVES

(2) Ivfiecion,Let P corr-espond to its tallg-entiftl point pi (see Ex. 2,

247). Here the proper united points are the inflexiolls,and cusps are such in a special sense. There is an(rn-2, n-2) correspondence, so that

a=rn-2, /3=11,-2 and y=2,since the line 0 meets f in two points at P.

If there are L inflexions and K cusps,L+K=(m-2) + (n-2)+2.2.p

=(m+n-4) +4p.Putting 2p=m-2n+2+K, we get L=3(m-n)+K. ( 14.9).

In the case of a unicursal curve, we have the number ofunited points equal to 'K=L+K=rn+n-4

L=m+n-4-KEx. A conic may have a five-pointic contact at any point P of a

cubic. ThIS conic therefore meets the cubic in another point P'.Between P and P' there is theu a (I, co) , correspondence, where w isthe number of conics drawn through P' having five-pointic contactelsewhere.

The united points of this correspondence will be those wherea conic has a aix-pointic contact. These are called ' < seetnctic " points.

Now, in the case of a non-singular cubic curve, the number ofsextactic points, as we shall show later on, is 27. Thus with the helpof the correspondence formula we may find the value of w. Herc p= 1.

:.27=1+w+2-5.1 01' W= lG .We thus obtain the theorem:Throuqh. any point of a non-8ing1tla,' cnbic "'illleen conics call be drawn

which will. hat'e fire-pointie contact icitt. the wbic eiseuhcrc,

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Ganguli, Theory of Plane Curves, Chapter 12

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