Or ELLIPTIC FUNCTIONS. 
447 
68. It is to be remarked that (3 , q being always given linearly in terms of it is the 
same thing whether we seek in this manner for the values of (3, q or for that of ^ ; but 
the latter course is practically more convenient. Thus in the cases n— 5, n = 7 we 
require only the value of 
In the case n= 11, where the coefficients are cc, (3, y, s, £, it has been seen that y, § 
are given as cubic functions of ~ : seeking for them directly their values would (if the 
process be practicable) be obtained in a better form, viz. instead of the denominator 
(F'y) 3 there would be only the denominator F'('y). 
69. I consider for ^ the cases n— 3 and 5 : 
w=3,/=0, 1, 2, 3, then ^=0, 3, 6, 1 ; 
and we write down the equations 
S M — A 
s m= a “‘ 
S : m=A'«, 
S M=° 
S M= 0; 
viz. if we had in the first instance assumed S^=A-j-Bw 8 + 
S ^=A« 4 +Bw _4 -f- . . , whence B and the succeeding coefficients all vanish ; and so in 
other cases. We have here only the coefficients A, A' ; and these can be obtained without 
the aid of the ^-formulae by the consideration that for u = 1 the corresponding values of 
M 
v=l, - 1 , - 1 , - 1 , 
~=3, -1, -1, -1, 
whence A=0, A' =6 ; or we have the equations 
Sm=0, S^=6 W 3 , S^=0 
M 
M' 
giving as before 
S$=6«, 
(2v 3 + 3 v 2 u —u)^= 3 (vht ? + 2 u 5 v + 1 ) u. 
reducible by means of the modular equation to 
70. n— 5. Corresponding to < /’=0, 1, 2, 3, 4, 5, we have ^=0, 5, 2, 7, 4, 1, 
3 N 2 
