45] 
ON THE THEORY OF ELLIPTIC FUNCTIONS. 
299 
The case corresponding to the denominator in the multiplication of elliptic func 
tions is that of G 0 — 1, Gi = 0. It is easy to form the table'— 
G 0 = 1, 
G,= 0, 
C 2 = — 2 n (ft — 1), 
C s = 8 n (ft — 1) (n — 4) a, 
(7 4 = — 4 n (n — 1) (n — 4) [n + 75] 
— 32n (w — 1) (w — 4) (n — 9) a 2 , 
(7 5 = 96 n (n — 1) (n — 4) (n — 9) [ft,+ 44] a 
+ 128 n (n — 1) (ft, — 4) (ft — 9) (ft, — 16) a 3 , 
<7 6 = - 24 n (n -1) (n - 4) (ft - 9) [17n 2 + 403w + 9000] 
— 960 n (n — 1) (n — 4) (ft, — 9) (ft — 16) [ft + 41] a 2 
— 512 n (n — 1) (ft — 4) (ft, — 9) (ft — 16) (ft — 25) a 4 , 
G 7 = + 96 ft (ft — 1) (ft — 4) (ft — 9) (ft — 16) [79ft 2 + 2825ft + 36180] a 
+ 7168 % (ft — 1) (ft — 4) (ft — 9) (w — 16) (ft — 25) [ft + 42] a 3 
+ 2048 ft (ft — 1) (ft — 4) (ft — 9) (ft — 16) (ft — 25) (n — 36) a 5 , 
G % = — 48 ft (ft — 1) (ft — 4) (ft - 9) [283ft 4 - 26978ft 3 + 277827ft 2 - 5491932ft + 127764000] 
— 3840 ft (w - 1) (ft - 4) (ft - 9) (ft - 16) (n - 25) [23ft 2 + 1069ft + 23436] a 2 
— 15360 ft (ft — 1) (ft - 4) (ft - 9) (ft — 16) (ft — 25) (ft — 36) [3ft + 133] a 4 
— 8192 ft (ft — 1) {n — 4) (ft — 9) (ft —16) (ft — 25) (ft — 36) (ft — 49) a 6 , 
&c. 
in which of course the coefficient of the highest power of n, in the successive co 
efficients G r , is the value of C r obtained from the equation (8). With regard to the 
law of these coefficients I have found that 
G r = (-) r+1 2 2r ~ 3 ft (ft - l 2 ) ... [ft - (r - l) 2 } 6V a»- 2 
+ 2 2r_G ?i (ft — l 2 ) ... [ft — (r — 2) 2 } G r ~ a r-4 
+ 2 2r-9 ft (ft — l 2 ) ... {ft- — (r — 3) 2 ] C7 3 a’’ -6 
+ &c. 
(where however the next term does not contain, as would at first sight be supposed, 
the factor ft (ft — l 2 ) ... {w —(r —4) 2 }). And then 
Cr 1 = 1, 
cy = (r - 3) [ft (2r - 7) + (r -1) (8r - 7)], 
Cy=|(r — 4)0—5) [ ft 2 (4r 2 — 24r + 51) 
+ ft (32r 3 — 220r 2 + 412r — 255) 
+ 2 (r — 1) (?" — 2) (32r 2 — 88?' + 51)]. 
38—2