OP ELLIPTIC FUNCTIONS. 
421 
The values are in fact given as transcendental functions of q; viz. denoting by 
M 0 , M n Mo . . ., M„ the values corresponding to v 0 , v„ v 2 , . . . v n respectively, and writing 
(1 + ?) (1 + g 3 ) (1 + 5 g) . . . (1 - «? 2 ) (1 - g 4 ) (1 ~ q t; ) ■ • • 
m) ~ (1 ■ - q ) (1 - ? 3 ) (1 ~ ? 5 ) • • • (1 + ? 2 ) (1 + $ 4 ) (1 + 4 *) • • • 
=l + 2q + 2q* + 2q° + 2q' 6 +..., 
then we have 
M ( 
L~i 
n 
> 2 (g) . 
?*(?”) 
M,= 
fig) 
1 5 
f{*q n ) 
(a an imaginary nth root of unity) 
M,= A' 
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 n=0 shall be M = l. 
The equations are : — 
1 
M 4 
n=0, this is 
1 
+ M 3 * 
0 
CO 
+ 
rH 
1 
+ M 2 ‘ “ 
-6 
u— 1, it is 
+M- 8(l-2«») 
(m +1 ) (m~ 3 ) 
o 
II 
CO 
M 6 
n=0 or 1, this is 
+M 5 * 
-10 
1 
l-J 
1 
Cn 
. 1 
+ 35 
+M 3 - 
-60 
+M 2 ' 
+ 55 
. -26+256n 8 (l-0 
+5 = 0. 
MDCCCLXXIV. 
3 K 
