74 
A THEOREM IN ELLIPTIC FUNCTIONS. 
[728 
In fact, writing herein a + 7 = 0, that is, 7 = — a, the right-hand side becomes = 0 ; 
and the arcs on the left-hand side are a + /3, a — ¡3, — a + S, —a—S, which represent 
any four arcs the sum of which is = 0. 
Writing in the last-mentioned equation x, y, z, w for the sn’s of a, (3, 7, $ 
respectively, 
also 
P = X 2 — y 2 , 
Pj = z 2 — w 2 , 
Q = 1 — x 2 — y 2 + k 2 x 2 y 2 , 
Q 1 = 1 — z 2 — w 2 + k 2 z 2 w 2 , 
R = 1 — k 2 x 2 — k 2 y 2 + k 2 x 2 y 2 , 
Rj — 1 — k 2 z 2 — k 2 w 2 + k 2 z 2 w 2 , 
D = 1 — k 2 x 2 y 2 , 
D 1 = 1 — k 2 z 2 w 2 , 
the equation 
is 
j^PPi.QQi 1 RRi 
DDj + DD, k 2 DD 1 
k' 2 2k' 2 (x 2 — z 2 ) (y 2 — w 2 ) 
that is, 
k 2 DD l 
1 k' 2 
- k' 2 PP x + QQ } - ^ RR, + ^PP 1 + 2 k' 2 (a; 2 - z 2 ) (y 2 - tv 2 ) = 0. 
It is easy to verify that the terms of the orders 0, 1, 2, 8 and 4 in x 2 , y 2 , z 2 , w 2 
separately destroy each other; for instance, for the terms of the order 2, we have 
— k' 2 (x 2 — y 2 ) (z 2 — w 2 ) + {{x 2 + y 2 ) (z 2 -f w 2 ) + k 2 (x 2 y 2 + z 2 w 2 )) 
— ^2 W + 2/ 2 ) + w ~) + h 2 (x 2 y 2 + z 2 w 2 )} 
k' 2 
+ -- {— k 2 (x 2 y 2 + z 2 w 2 )] + 2k' 2 (x 2 — z 2 ) (y 2 — w 2 ) = 0, 
that is, 
— k' 2 (x 2 — y 2 ) (z 2 — w 2 ) + (1 — k 2 ) (x 2 + y 2 ) (z 2 + w 2 ) 
+ (k 2 — 1 — k' 2 ) (x 2 y 2 + z 2 w 2 ) + 2k' 2 (x 2 — z 2 ) (y 2 — w 2 ) = 0 ; 
or, omitting the factor k' 2 , this is 
— (x 2 — y 2 ) (z 2 — w 2 ) + (x 2 + y 2 ) (z 2 + w 2 ) — 2 (x 2 y 2 -f z 2 w 2 ) + 2 (x 2 — z 2 ) (y 2 — w 2 ) = 0, 
as it should be. 
The theorem in its original form was obtained by me as follows: using the elliptic 
coordinates p, q, r, such that 
X 2 y 2 z 2 
b 1 = 1 
a+p b+p c+p ’ 
x 2 y 2 z 2 
a + q b + q c + q 
= 1, 
X 2 y 2 z 2 
1—y. 1_ _—_ 
a + r b + r c+r