OF ELLIPTIC FUNCTIONS. 
423 
The Multiplier as a rational function of u, v . — Article Nos. 30 to 36. 
30. The multiplier M, as having a single value corresponding to each value of v, is 
necessarily a rational function of it, v ; and such an expression of M can, as remarked by 
Konigsbekgek, be deduced from the multiplier equation by means of Jacobi’s theorem, 
1 a(1 a 2 ) dk . 
iVi —nk(l-k*)d\’ 
viz. substituting for Jc, X their values u 8 , v 8 , and observing that if the modular equation 
be F(a, v)=0, then the value of ^ is =— F(v)-^-F(w), this is 
___1 (l-y>Fw . 
n (1 — m 8 )mF'm * 
and then in the multiplier equation separating the terms which contain the odd and 
even powers, and writing it in the form <b(M 2 )+MT r (M 2 ) = 0, this equation, substituting 
therein for M 2 its value, gives the value of M rationally. 
The rational expression of M in terms of u, v is of course indeterminate, since its form 
maybe modified in any manner by means of the equation F [u, w) = 0 ; and in the expres- 
sion obtained as above, the orders of the numerator and denominator are far too high. 
A different form may be obtained as follows : for greater convenience I seek for the value 
not of M but of 
31. Denoting, as above, by M 0 , M,, . . . M„ the values which correspond to v 0 , v 1} . . . v n 
respectively, and writing S + • • • +jjj- &c -> we have S^-, SA-, &c., all of 
them expressible as determinate functions of u ; and we have moreover the theorem that 
each of these is a rational and integral function of u : we have thus the series of equations 
S S= A > Sg=B,...,sg=H, 
where A, B, . . . H are rational and integral functions of u. These give linearly the 
different values of ; we have in fact 
(v 0 — VJ . . . (v 0 — v n ) |y-=H— GS^+FS^Uj . . . + Av{o a . . . v n , 
where Sw n Sv^, &c. denote the combinations formed with the roots v lt v 2 , . . . v„ (these can 
be expressed in terms of the single root v 0 ); and we have also (v 0 —v x )... (v 0 — v n )=T'(v 0 ) : 
the resulting equation is consequently FT 0 ^ =H(u, v 0 ), R a determinate rational and 
integral function of (u, v 0 ) ; but as the same formula exists for each root of the modular 
equation, we may herein write M, v in place of M 0 , v 0 ; and the formula thus is 
FT.^=R(w, v), 
3x2 
