252 Proceedings of the Royal Society of Edinburgh. [Sess. 
As I involves functions of an imaginary variable v, we must transform 
the integrals to functions of a real variable in order to calculate the values 
of the wave-lengths of the lines which arise. 
Since 
h^sn^-‘{v, k) = 
sn^{v, k) = 
1-^2 
l + e2 
1-^2 
Putting v = iw and using the results 
j. .sniiv.k') . Il + e^ 
suiiw, k) = i — > ’ / = t\ 
cn{tv, k ) A 1 - 
cn{iio, k) = ^ , 
C7l{lV, k ) 
dn{iiv^ k) = dn{iv, k')lcn('W, k ') , 
we have 
sn^{w, //) = 
1 
2k'‘^ ’ 
cn^{w, k') = (1 - 2A;2)/2A^'2^ 
div^iio, k') = 1/2 . 
c?^(r, k) m{io, k') 
r^/2{l-2k^). 
Also 
S7i(v, k)d7i{y, k) i S7i(w, k')d7i{w, k') i 
i 7j(v) i Z{iiu) 
snlw.k') -j , Tviv r// 
^ ’ -ld7i{wk ) + , + Z(w;, k ) 
C7l{n), k') 
1 
2KK' 
■rnv 
k ') . 
cn V 
-y/2( 1-2*2) 2KK' 
k 
1 = 
E - K + K - V2(l - 2*2 )/ + KZ(w, *')) ] • 
From (7) we have 
71^ 
+ 7^2 A;2 
_A/2 
~k^ 
Since 
we have 
(8) . 
+ 71^ Kr 
1-V2(1-2*2)||^ + ZK*')| 
fK„, ,,, /, KE' K KE'\n 
V2(l - 2*2) I ^ EK * ) + J . 
7t/2 = K'E + KE'-K1v', 
1 - V2(l-2/,;2)|^y(!<’, *') - E (», *') ^ + E(?ff, *')] 
