Ellliptie Integral of the Second Kind. 507 
whence 
K cos 20—K’ sin 20 
TT 
= —sin 0+ 30 i 70 pests 110+& 
=— seein ors qzsin +or 2. gz sin + WC. 
Thus 
2H _K 1 ges Bh oS, 
— == +58 in 30-+ 5 SFT (mae Ez ~; sin 116+ &e.; 
or, substituting for = its value, 
yA i gs: Me 
— =sin 645 sin 30+ 5p sin bes sin 70 
T P74 
2 2 2 
-- aoe = sin 90+ 5E. sia sin 116+ 92. ees S sin 136+ &c., 
which is the series for E corresponding to (1). 
Formule for K, E, 1, U, V, W, §§ 5-9. 
§ 5. The results obtained in the previous sections form 
part of a general system of formule which may be stated as 
follows :— 
Consider two quantities I and G defined by the equations 
I=EK-K, 
G=H—k’K ; 
and let U, V, W denote 
4+G), 3(G+H), $(H+)) 
respectively. 
The values of the six quantities, expressed in terms of HE 
and K, and of I and K, are therefore 
H=H =I+K, 
I =HK-K =, 
G=H—k?K =1+?K, 
U=E—-4(14+k”)K =14+4/K, 
V=H—-$k?K =1+3(1+/)K, 
W=E-1K =I+iK. 
