Elliptic Integral of the Second Kind. 509 
2W’ 1 ah Sits 
ee. 
—— — 5 008 30— ; 97 4 088 70— 5E ge > 
cos 110—&ce., 
AS = sin @— = sin 30— 5 a qisin 10— oF ae sys Sin 110—&e., 
/ 
si oF 5-5 COS 110+ée., 
i gee 
Be 10+ 2.6 
§ 8. It appears from § 5 that the quantities I’ and V’ 
resemble K’ in being transformable, by the change of modulus 
from k’ to e—**, into expressions of the form e+%(P’+7P), 
where P’ is the same function of k” that P is of #?. Perhaps 
the simplest method of obtaining the sine-series for E is by 
deducing the sine-series for W from the i/-series for I’ and 
then adding the sine-series for $K ; and it was in this manner 
that I was first led to the series for E*. It will be noticed 
that the process followed in § 4 also gives in the first instance 
the sine-series for HH —4K or W, and that the series for E is 
deduced from it by the addition of that for 4K. 
By the change of k’ into e-?”, 
kK’ becomes $e—(K’—iK), 
=cos 0+ 52008 30+ 
and therefore 
V’—k’K’ becomes e-®( I’ +7E). 
if then we start with the 4’-series for I’/—#’K’ and transform 
k’ into e~*, we obtain directly the sine-series for E by equa- 
ting the imaginary parts in_the resulting formula. The 
transformation therefore (of k’ into e~2) which converts 
K’ into de(K/—iK) 
converts also 
K/—EH’+2#’K’ into e~®(K’—H'—iE) ; 
and we may therefore derive the sine-series for H from the 
k’-series for K’— Hi’ + k’K’ by a process exactly analogous to 
that by which the sine-series for K is derivable from the 
k’-series for K’. 
The six quantities H, I, G, U, V, W are considered in detail 
in the Quarterly Journal of Mathematics, vol. xx. pp. 313- 
361, and in the Proceedings of the Cambridge Philosophical 
Society, vol. v. pp. 184-208. The formule in these papers 
show that E forms one of a triad of corresponding funda- 
mental quantities of which the other two members are Land G. 
§ 9. It may be remarked that I=—J, where J is the 
* Proc. Camb. Phil. Soe. vol. v. p. 204. 
Phil. Mag. 8. 5. Vol. 19. No. 121. June 1885. 2M 
