364 
[57 
57. 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 50—51.] 
We have seen [45] that the equation 
dz 
n (n — 1) x 2 z + (n — 1) (ax — 2x 3 ) + (1 
o d 2 z a . „ ..dz _ 
ax 1 + x 4 ) — 2n (a- — 4) ^ = 0 
da 
is integrable, in the case of n an odd number, in the form z = B 0 + B^ 2 ... + B± {n - 1 )X n ~ 1 ; 
and the coefficients at the beginning of the series have already been determined ; to 
find those at the end of it, the most convenient mode of writing the series will be 
: +£ 
i-YD r 
1.2 ... (2r + 1) 
and then the coefficients D r are determined by 
rlT) 
D r+2 = (2 r + 3) (n - 2r - 3) D r+1 a-2 n (a 2 - 4) 
- (2 r + 3) (2 r + 2) (n -2r-2)(n-2r- 1) D r . 
The first coefficients then are 
A = i, 
A = (n - 1) 
D-2 — 2 (n — 1) (n + 6) + (n — 1) (n — 9) a 2 , 
D 3 = 6 (n — 1) (n — 9) (n + 10) a + (n — 1) (n — 9) (n — 25) a 3 , 
D 4 = _ 36 (n - 1) (n 3 - 13n 2 + 36n + 420) 
+ 12 (n — 1) (n— 9) (n — 25) (n + 14) a 2 
+ (n — 1) (n — 9) (n — 25) (n — 49) a 4 ,