140 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578 
• + ^ 7 . + 3168(1 — 2M 8 ) 
+ i . - 4620 - 3. II 2 .256m 8 (1 - O 
+ i • 4752+11 • 4096 “ 8 ^ ” w8 )i “ 2w8 ) 
+ -^- 4 . — 3465 — 3.7.11.512m 8 (1 — w 8 ) 
+ № *^ +1760 +11 ' 83 ' 2048w8 ^“ w8 )l( X ~ 2m8 ) 
+ ^- 2 . — 594 — 9.11.37.256it 8 (l — m 8 ) — 3.11. 131072 {m 8 (1 - m 8 )} 2 
+ ^.{l20 + 15.4096w 8 (1 - m 8 ) - 524288 [m 8 (1 - it 8 )} 2 } (1 - 2m 8 ) 
-11=0. 
The Multiplier as a rational function of u, v. Art. 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 u, v; and such an expression of M can, as remarked 
by Königsberger, be deduced from the multiplier equation by means of Jacobi’s 
theorem, 
... .1 X(l-V) <&-. 
n k (l — & 2 ) d\ ’ 
viz. substituting for k, \ their values m 8 , u 8 , and observing that if the modular equation 
be F(u, v) = 0 so that the value of is = — F'(v) -r- F' (u), this is 
a/2 = _ 1 (1 -F)vF’v _ 
n (1 — u s ) uF'u ’ 
and then in the multiplier equation separating the terms which contain the odd and 
even powers, and writing it in the form <f> (ili 2 ) + ilf'F (M-) = 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 may be modified in any manner by means of the equation F(u, v) = 0; and in 
the expression obtained as above, the orders of the numerator and the 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