OF ELLIPTIC FUNCTIONS. 
429 
+ 4w 24 x'° 
- 2 u 2i x 11 
- if* 
Term in { } lias factor 
1+ £«+-*■; 
1 a 1 V ’ 
«=1, term in { } is 
= (l+tf)'(l-+) 5 . 
+ 8m 8 ( 8+ 57m 8 + 46m 16 )# 10 
+56m i6 (2+m 8 ) 
X u 
— 4m i6 ( 56 + 161 m 8 + 
56m'> 12 
+56m 24 (1+2m 8 
) x' 3 
+ 8 m 24 ( 46+ 57m 8 + 
8u ' 6 ) x u 
- 8z* 24 ( 16+ 25 u 8 + 
4m ,6 > 15 
- m 24 ( 16+305m 8 +144m 1( > 16 
+ 4m 24 ( 8+ 51m 8 + 
1 Qu l6 )x' 7 
+ 28m 32 ( 1+ 4m 8 ) 
X 18 
-28m 32 ( 2+ 3w 8 ) 
x' 9 
— 14m 40 
X 20 
+ 4m 40 (7 + 2m 8 ) 
X 21 
- 4?^ 48 
X 22 
- 4m 48 
X 23 
+ M 48 
X 24 } 2 
Term in { } has factor 
1 + -# + ~ # 2 + — x 3 ; 
1 a 1 a v 
(+) 
u— 1, term in { } is (l-f-#) 10 (l— x) 14 . 
The transformations n= 3, 5, 7, 11. — Article Nos. 41 to 51. 
41. The cubic transformation, n= 3. 
I reproduce the results already obtained ; since there are only two coefficients a, /3, 
these are also the last but one and last coefficient §, <r. Hence, from the values of a, (3, 
§, <r, we have 
a = 1, 
v s ( 1 u 4 
2 ci -u\M~V 4 
2 3=M-! 
e= : 
1 1 w 
the two values of ^ are thus =1H — — , giving the modular equation 
v 4 + 2«V — 2 y m — u 4 = 0 ; 
and we then have 
1 — y 1— x /v— u 3 x \ 2 
1+?/ l+a?yy+w 3 #y 
3 L 
MDCCCLXXIV. 
