166 A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [578 
^where, if f= or < 4, there must be on the right-hand side no negative power of u ; 
but if f> 4, then the highest negative power must be and 
8v* +f ™ = A'u n f +i + B'u n f~ 4 + ..., 
where on the right-hand side there must be no negative power of u. 
68. It is to be remarked that /3, p being always given linearly in terms of ^ 
it is the same thing whether we seek in this manner for the values of /3, p or for 
that of but the latter course is practically more convenient. Thus in the cases 
n — 5, w = 7 we require only the value of 
In the case n = ll, where the coefficients are a, /3, y, 8, e, f, it has been seen that 
y, 8 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'v) 3 there would be only the denominator F'(v). 
69. I consider for -L the cases n = 3 and 5: 
/1=3, /=0, 1, 2, 3, then p = 0, 3, 6, 1; 
and we write down the equations 
1 v 4 
S M =A ’ giving S m= A < 
M 
S^A'u, 
S M~ 0, 
S M~°’ 
viz. if we had in the first instance assumed S A +Bu s +.., this would have given 
v 4 
8Au 4 + Bu~ 4 +.., 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 g-formulse by the consideration that for u = 1 the corresponding 
values of v, ^ are 
» =1, -1, -1, -1, 
1 
~M 
= 3, -1, -1, -1,