472 Mr Dixon, Jacobi's double-residue theorem 



Jacobi's double-residue theorem in relation to the theory of 

 point-groups. By A. C. Dixon, Sc.D,, F.R.S., Trinity College. 



[Read 23 Mai/, 1910.] 



In this paper I have shewn how Jacobi's theorem leads 

 directly to the chief general propositions of the theory of point- 

 groups in a plane, and have also given a discussion of a converse 

 theorem. No account has been taken of coincidences among the 

 points of a group. 



1. Jacobi's theorem is as follows. Let u, v, w be three 

 polynomials in two variables oc, y, of degrees m, n, m-\-n — S 



respectively, and let J be the Jacobian ^^-^ — ^ . If the mn points 



o [x, y) 



of intersection of the curves w = 0, v = are all distinct and at 

 finite distance from the origin, and are denoted by {x^, y^) (r=l, 

 2 . . . mn), then 



vnn 



^ w(Xr, yr)/JiXr, yr) = (1). 



r=l 



(See for instance Netto, Vorlesungen ilber Algebra, vol. ii, 

 pp. 165—173.) 



The following proof is a modification of one given by Netto, 

 after Kronecker. 



If w is an arbitrary polynomial of degree m + n— 3, it contains 

 I (m + w — 1) (m + ?i— 2) arbitrary coefficients, and if w is re- 

 stricted by being supposed to vanish at (x^, yr) {r=l, 2...mn), 

 this number of coefficients is brought down to 



^ {m + n — 1) (m + n— 2) — mn 



or ^(m- 1) (m - 2) -\- ^(n -1) (n~2) -1 



if all the conditions w (x^, yr) = 0, to be satisfied by those co- 

 efficients, are independent, that is, unless there is some relation 



S Ar W {Xr, yr) = 



satisfied by all polynomials w of the degree m + n— S. 



Now if <j), yfr are any polynomials of the degrees n — S,m — S, 



Ucfi + V^lr 



is of the degree m + n — S and vanishes at the mn points, and 

 contains ^ (m—l) (m — 2) + ^ (n — 1) (n — 2) arbitrary coefficients, 

 namely those in ^ and yjr, these being all effective unless for some 

 set of coefficients u(fi + v-yjr is identically zero, that is, unless u, v 



