858] 
425 
858. 
COMPARISON OF THE WEIERSTRASSIAN AND JACOBIAN 
ELLIPTIC FUNCTIONS. 
[From the Messenger of Mathematics, vol. xvi. (1887), pp. 129—132.] 
The Weierstrassian function a (u) corresponds of course with Jacobi’s H (u), but 
it is worth while to establish the actual formulae of transformation. 
Writing, for a moment, 
CO = <»! 4- Wx, 
CO = co 2 + iv 2 , 
it is convenient to assume co 1 v 2 — co 2 v l positive; we then have 
2 (ijco' — r) co) = 4-7ri; 
in particular, this will be the case if co = co 1} co — iv 2 , where co 1} v 2 are each positive. 
To reduce the periods into the Jacobian form, we may assume 
CO = \K, 
co' = XiK', 
(where observe that, if as usual k, K, K' are each real and positive, and if as above 
co = coco' = iv 2 , and v 2 positive, then also X, will be real and positive). We have 
iK' co' ^ 
K = ~co’ or sa y ? = e =e " , 
ay 
which determines first q, and thence k, K, K', as functions of —, and we then have 
CO 
X = ^ > either of which equations gives X as a function of co, co'. Conversely, 
starting with k, X, the original equations give the values of co, co'; those of rj, f will 
be determined presently. 
C. XII. 
54