291 
45] ON THE THEORY OF ELLIPTIC FUNCTIONS, 
or writing — u for u and subtracting, yjr^u being an even function, 
2Au = iri — (iK' — u) + k 2 ^ // (iK' + u), 
or putting u = K, 
2AK = iri — (iK 7 — K) + lc 2 'yjr // (iK' + K). 
Now sn 2 (u + K) — sn 2 (u — K) = 0, 
and therefore \Jr / (u + K) — ■\fr / ( u — K) = 2 , 
// ( w + #) - ir„ ( w -K) = 2u ; 
or O'iT + if)~ ^ (iZ 7 - K) = 2iK'±K. 
Also i? (u) = u — 
or 
E = K-k^K, i.e. = 
Hence 
¿-urii-£)+*, 
log sn xi = kFJr lf u - kFf-,, (u + iK’) + uiK’ 0 - ^ + B 
= ***„« - k^„ (u + iK') + i [(« + iKy - vF] (l - |) + 
i.e. 
log sn m = log © (u + iK') — log ©w + + B', 
or, changing the constant, 
sn u 
_ C( M-®(u + iK') 
©m 
Now, to determine (7, write u — iK’ for u; this gives 
1 _ r( M {u - iK ' ] ©w 
k sn u © (u - iK) ‘ 
and again changing n into — u, 
— sn u 
_ Cc S®(u-iK') : 
&u 
whence, multiplying these last two equations, 
_ TtK' 
k 
+ B\ 
37—2