138 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
[578 
578] 
The Multiplier Equation. Art. No. 29. 
29. The theory is in many respects analogous to that of the modular equation. 
To each value of v there corresponds a single value of M; hence M, or what is the 
prime, the order is = n + 1. The last term of the equation is constant, and the other 
coefficients are rational and integral functions of u s , of a degree not exceeding % (n — 1); 
and not only so, but they are, n = 1 (mod. 4), rational and integral functions of w 8 (l — u s ), 
and n = 3 (mod. 4), alternately of this form and of the same form multiplied by the 
factor (1 — 2ii 8 ). 
The values are in fact given as transcendental functions of q; viz. denoting by 
M 0 , M u M 2 , ..., M n the values corresponding to v 0 , v 1} v 2 , ... , v n respectively, and writing 
, _ (1 + qH1 + 9 3 ) (! + g 5 )-(l - f) (1 - 9 4 ) (1 - fh. 
* W (1 - q) (1 - ? 3 ) (1 - <f) ... (1 + f) (1 + q*) (1 + q*) . 
= 1 -f 2q + 2q 4 4 2q 9 + 2q ls + ..., 
then we have 
Hence, the form of the equation being known, the values of the numerical coefficients 
may be calculated; and it was in this way that Joubert obtained the following results. 
I have in some cases changed the sign of Joubert’s multiplier, so that in every case 
the value corresponding to u = 0 shall be M=l. 
The equations are: 
u = 0, this is 
u — 1, it is 
-3 = 0.