298 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
[45 
relation G r x = 1, since the arbitrary constants of the integration affect only the terms 
multiplied by 2 r , S r , &c. 
Assume G r p 
and substituting this value, 
(1 -p)L p =(1 -p){LP~ l 
(1 -p)M p - 2\p (2 - 2p) L p = (1 - p) {Mp- 1 - IILp- 1 }, 
(1 - p) Np - 2p (3 - 2p) Mp = (1 -p) [Np~ x - IMP- 1 + 12LP- 1 }, 
(1 -p)QP - 2p (4 - 2p) Np = (1 -p) [Op- 1 - SNp~ x + 30Mp- 1 } , 
the law of which is obvious, the coefficients on the second side in the çth line being 
1, 4^ — 19, and (2q — 3) (2q — 2) respectively. By successive integrations and substitutions 
Lp - Lp- 1 = 0, 
Mp - Mp- 1 = 4p- 11, 
Np - Np- 1 = - 8p 3 + 2 6p 2 + 4>9p - 114 ; 
Lp = 1, 
MP = (p- 1) (2p - 7), 
(the constants determined by M 1 = 0, N 1 = 0, 0 2 = 0, P 2 — 0, ... so as to make C r p con 
tain positive powers only of r). 
The following are a few of the complete values of C r p , the constants determined 
so as to satisfy C P+1 P = 0 (except CV = 1), and the factorials being partially developed 
in powers of r, viz. 
Gp = (r - 3) (2r - 7), 
Gp — 1 (r — 4) (r — 5) (4r 2 — 24r +51), 
CP = i{(r- 5) (r - 6) (r - 7) (8r 3 - 60r 2 + 286r + 63) + 384 (9r 2 - 93r + 242 - 2.4 1 '- 5 )}, 
&c. 
(it is curious that Gp, Gp, Gp, all three of them vanish). It seems hopeless to con 
tinue this investigation any further. 
Returning to the equation (8), and assuming for z an expression of the same 
form as before, we have, corresponding to the equations before found for the co 
efficients C r , 
dC 
G r+2 = — (2r + 1) (2r + 2) (n — 2r) (n — 2r — 1) G r — (2r + 2) (n — 2r — 2) aG r+1 + 2n (a 2 — 4) .