250 
[754 
754. 
ON THE CONNEXION OF CERTAIN FORMULÆ IN ELLIPTIC 
FUNCTIONS. 
[From the Messenger of Mathematics, vol. ix. (1880), pp. 23—25.] 
In reference to a like question in the theory of the double ^-functions, it is 
interesting to show that (if not completely, at least very nearly) the single formula 
that is, 
n (u a)-u &a I tlor 0(M - g) 
n ’ ' 0£t 2 “0(» + «)’ 
k 2 sn a en a dn a sn 2 u du ©A .. ©(w — a) 
= “0ï + i lo «ë(ST5) ; 
1 — k 2 sn 2 a sn 2 u 
leads not only to the relation 
. ^ , 2k'K . , - 
log ®u = \ log + i (1 
— v? — k 2 J du J du sn 2 u, 
between the functions ©, sn, but also to the addition-equation for the function sn. 
Writing in the equation a indefinitely small, and assuming only that sna, cnct, 
dn a then become a, 1, 1, respectively, the equation is 
r i „ 7 a© 0 . , ©îî — a © a 
k-a sn 2 « du = u „ _ + log „—— 
J 0 ©0 2 6 ©« + «© « 
©« — aWu 
©"0 ©'« 
= ua — « FT" , 
©0 ©t£ 
that is, 
©"0 [ 2 
•pr- = u -prpr — k 2 du sn- u, 
©M ©0 Jo 
or, integrating from u = 0, this is 
©"0 f f 
log ©it = C+^id — k 2 du du sn 2 u, 
©0 Jo Jo