474 il/r Dixon, Jacohi's double-residue theorem 



Since ccPyi - ^PtjI = (xv - ^p) yi + {if - 7]i) p, the sum of two 

 terms Avhich contain the factors x — ^, y — t] respectively, we have, 

 taking the terms of u separately, 



u (x, y)-u (e 7?)= U, {x-^)+U, (y-v) (3) 



where Ui, U« are polynomials of degree m — 1 in x, y, f, t), and 

 when x=^ and y = r], Ui, C/g are equal to the derivatives Ui, u^- 

 Similarly 



v(a^,y)-v(^,v)=V,(x-^)+V,(y-v) (4) 



where F^, F^ are of the degiee n— 1 in x, y, ^, rj and reduce to 

 the derivatives ■Vj, Vo when x = ^, y = 1]. 



Let Ui V.,— t/g Fj = A (x, y, ^, ?;), then, by substituting 



^r, y,-, ^s, Vs for X, y, f, t; in (8) (4) 



we find H^ {xr, yr, Xg, yg) — 0, when r=f^s, while ii =J {x,., y,), 

 when r = s. J{xr,yr) is not zero since the curves t< = 0, v = 

 have only isolated intersections. 



Now A {x, y, Xg, ys) — A (x, y, xt, yt) is of the degree m + n — '^ 

 only in x, y, the terms of degree m + ?i — 2 destroying each other. 

 Hence 



mn 



S A,. {A {xr, yr, Xg, ys) - A (x,., y,., Xt, yt)] = 0, 

 ,.=1 



that is AgJ(xs, y^) - AtJ(xt, yt) = 0. 



This holds for all suffixes s, t and therefore the relation (2) is 



tw(x.,., yr)/J(i^r,yr) = (1), 



which is Jacobi's theorem. 



2. It follows directly that if w vanishes at mn — 1 of the 

 intersections of u, v it vanishes at all, or that any curve of degree 

 (?n + n — 3) through mn — 1 of the intersections of two curves of 

 degrees m, n passes through all their intersections. 



Again, let (f) be an r'" (r <m + n — 8) vanishing at the first 

 mn — a of the intersections, and -yjr an arbitrary (m + n — r — 3)^*^. 

 We may put ^i/r for w and thus we have 



mn 



t 4> {Xr , 2/,.) y^r {Xr ,yr)/J (^r , ^r) = 0, 



r=mn—a+l 



an equation which includes 



^ (m + n—r—l)(m+n — r—2) equations, 

 linear in (f) {x^, y^) (r = mn — a + 1, ... mn), 



and therefore gives ^ (x.,., yr) = for all these values of r if 

 a = ^ (m + n — r — 1) {in + n — r — 2), 



