FUNCTIONS. [578 
578] 
A MEMOIR ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 
117 
(u = Vk, v = \/X); but 
X to introduce the 
y the value of il is 
^uation between u, v, 
er between u 2 , v 2 ; and 
*er powers of u 4 , that 
)r say an il&-modular 
v; these i2&-forms for 
iy 
6. We may from the 1) equations eliminate the \ (n— 1) ratios oc : /3 : <y :... , 
thus obtaining an equation in H (involving of course the parameter k) which is the 
n&-modular equation above referred to; and then il denoting any root of this equation, 
the £(n + l) equations give a single value for the set of ratios a : (3 : y : S : ..., so 
that the ratio of the functions P, Q is determined, and consequently the value of y 
as given by the equation 
1 — y _ (1 — oc) (P — Qxf . _ x (P 2 + 2PQ + Q 2 x 2 ) 
1 + y (1 + x) (P + Qxf ’ ° r y P- + 2 PQx 2 + Q 2 x 2 
The entire problem thus depends on the solution of the system of ^(?i+l) equations, 
(P 2 + 2PQx 2 + Q 2 x 2 )* = (P 2 + 2PQ + Q 2 x 2 ). 
fyn-i 
1c^ 
,n—2 
br,«; 
; so that H signifying 
¡-¡j S3* ; or substituting 
! ), 
tie number of terms is 
adric functions of the 
ns, each of the form 
ients of P, Q, say of 
The flk-Modular Equations, n = 3, 5, 7, 11. Article No. 7. 
7. For convenience of reference, and to fix the ideas, I give these results, calculated, 
as above explained, from the standard or w-forms. 
fi 6 
i2 5 
il* 
O 3 
O 2 
n 
H° 
k 2 
k 
1 
il 4 
+ 1 
o 3 
-4 
n 2 
+ 6 
i2 
-4 
o° 
+ 1 
-4 
+ 8 
-4 
¥ 
¥ 
¥ 
= 0 
+ 1 
-16 
+ 10 
+ 15 
- 20 
+ 15 
+ 10 
- 16 
+ 1 
= 0 
n = 3 : 
0 = 1, we have — 4 (& — 1) 2 = 0. 
n = 5 
0 = 1, we have — 16 (k 2 — l) 2 = 0.