478 Mr DLvon, Jacohis double-residue theorem 



(2) The point (^, ?;) where the same Jacobian is equal to uv ; 



(3) The points (|, Y,.) where cc = ^, v = 0: at these the Jaco- 

 bian is equal to u {y — ri)v„\ 



(4) The points (X,., ?;) where y = ih ^' = and here the 

 Jacobian is equal to v {x — |) u^. 



Thus substituting x, y for ^, r) we have 



w , "^^ io{xr,y,) ^^ tu{x,Y,) 



+ - 7 w ^ r r/ X + 



uu ,.t 1 {x - Xr) (y - yr) J {Xr , j/r) ,-=1 ^ (x, Y,.) ( Y, - y) v.. {x, Y,) 



+ ,ti «i (X„ 2/) (X, - x) V (X,, y)-"""-^'^' 



and the degree of w may be anything up to m + n— 1. We have 

 to examine the third and fourth terms on the left in (7). As to 

 the third term, let tu/uv be reduced to partial fractions as a 

 function of y, x being treated as parametric. The denominators 

 of these fractions are y — Yr{r = 1, ... n) and the factors of u. 



The fraction whose denominator is y— Yr has for its nume- 

 rator w {x, Y.))/u {x, Y^) v., (x, Y,.) and the sum of these n fractions, 

 with sign changed, forms the third term in (7). Let their sum 

 be brought to a common denominator, v, the numerator will then 

 be of degree ?i — 1 in y, and its coefficients being symmetrical 

 in Fi, Fa... F,i will be rational functions of x, but in general 

 fractional: let P denote this numerator. Similarly the other 

 partial fractions will have a sum Q/u, where Q is of degree 7n — 1 

 in y, and its coefficients are rational in x. 



rru w P Q 



Thus — = — + :^ , 



uv V u 



lu = Pu -\- Qv : 



but this identity determines P, Q uniquely if their degrees are 

 71 — 1, m — 1 in y, unless u, v as functions of y have a common 

 factor, which is only true for special values of a;, namely x^jX., . . . x^n- 

 Similarly, if R, S are integral in x and of degrees n — 1, m — 1 

 and are such that 



w = Ru + Sv, 



the fourth term in (7) must be — S/u. 

 Hence (7) becomes 



uv T=l{ai-Xr){y-yr)J{Xr,yr) V U ^ ^' 



and if P is not integral its denominator is a function of x only, 

 and that of ^ is a function of y only. These denominators must 

 in fact be 11 {x — Xy) and 11 (y — _y,.). 



