45] 
ON THE THEORY OE ELLIPTIC FUNCTIONS. 
297 
In general, by leaving n indeterminate, and integrating in the form of a series 
arranged according to ascending powers of x 2 ; then, whenever n is a square number, 
the series terminates and gives the denominator of the corresponding formula of multi 
plication ; but the general form of the coefficients has not hitherto been discovered. 
cc 
By writing instead of x, and then making n infinite, the equation (8) takes the 
form 
(9): 
and it is worth while, before attempting the solution of the general case, to discuss this 
more simple one 1 . 
Assume 
then it is easy to obtain 
Cr+ 2 = - (2r + 1) (2r + 2)G r - (2r + 2) aC r+1 + 2 (a 2 - 4) -p 
dot 
The general form may be seen to be 
G r = (-) r+1 {2 3r-3 Or 1 oF~ 2 + 2 2r ~ 6 G 2 cf* + ...}, 
and then 
C r+1 P-pC r P = - r (2r - 1) C^p- 1 +16 (r + 2 - 2p) C r P~ l . 
The complete value of C r p (assuming C r ° = 0) is given by an equation of the form 
G r P = °C r P + l C r P 2 r + 2 G v p 3 r ... +P- 1 G r P f, 
where °C r p , 1 G r P, are algebraical functions of r of the degrees 2p — 2, 2p — 4, &c. 
respectively; but as I am not able completely to effect the integration, and my only 
object being to give an idea of the law of the successive terms, it will be sufficient to 
consider the first or algebraical term 0 G r p , which is determined by the same equation 
as C r p , and is moreover completely determined by this equation and the single additional 
1 Writing (/3+2) for a, and putting z = e~ d ' 2 p, this becomes 
p=ß*p-ßx^ + (8ß + 2p)&i 
and if p='EZ n ß n , 
from which the successive values of Z 0 , Z x , &c. might be calculated. 
C. 
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