476 Mr Dixon, Jacohi's double-residue theorem 



where /^„_i is an arbitrary (??? — l)''^ and so on. Then a form 

 for (f) is fm_i fr-m+i — fn-i fr-n+i '■ this is an ?•"■ vanishing when- 

 ever u, V both vanish, except at the iti + n — r—1 points where 



a; = l), fm+n—r—i = ^• 



3. Theorem of Riemann and Roch. Take any g points (J.) on 

 a curve u — 0, of degree in. Through these describe an ?i''= curve 

 V = (w > 771 — 8), cutting u = in mn — q other points {B). 



Let X be the number of arbitrary coefficients in an n**^ vanishing 

 at all the points 5, and yu. the number of arbitrary coefficients in 

 an (m - 3)''' vanishing at all the points A. Then shall 



X = -I (?) + 1) (/i + 2) — mn + 5 + ^14. 



For if 0, ^ are of degrees n, m — 3 we have 



2 (f)ylr/J=0. 



Suppose (^ to vanish at J5; then we have here \ relations 

 satisfied by the values of y^ a,t A, but among these relations 



•| {n - m +l)(n-m + 2) + l 



are illusory, namely those given by putting (f) = v or w;^^ where % 

 is of degree n — m. The number of relations is thus reduced to 



\-h {n - m + l){)i -vi + 2)-l, 



but it is not less than this, for the number of illusory equations 

 is the number of linearly independent n^'^^ (j) which vanish at all the 

 points A, B. Let ^ be any such, then by a suitable choice of the 

 constant a we can make <^ — av vanish at a new point P on u = 

 and therefore contain u' as a factor, u' being that factor of u 

 which vanishes at P. Thus (f} — av=wu', say. If u = u'u" ..., 

 the factors u', u" . . . being of course irreducible and of degrees 

 m, in", . . . then tu is of degree n — m and vanishes at the nm" 

 points where ^ = and i; = meet u" = 0. Hence lu must contain 

 u", and so on for all the other factors. Thus even when u is 

 composite, the only n^'^^ which vanish at all the points A, B are 

 included in the form av + ux- 



The values of an arbitrary {m — of'^ at the points A are then 

 connected by 



\-^{n-m+l) (n - m + 2) - 1 



linear relations, and fi, the number of arbitrary coefficients in an 

 (m — 3)*" through the q points A is therefore 



^ |(m - 1) (m -2)-q + \- ^{n- m + l) {n - m + 2) - 1, 

 or X + mn-^{n-{- l)(w+ 2)-q (5). 



