1884.] K', E', J', G' in powers of the modulus. 185 



K, K', 



E, E', 



J = K-E, J' = K' - E', 



G = E- k' 2 K, G' = E' - k 2 K', 



J-G = (l + Jc' 2 ) K -2E, J'-G' = (l + k 2 ) K' - 2E, 



E + G=2E- k' 2 K, E'+G' = 2E' - k 2 K', 



E-J=2E-K, E'-J' = 2E' -K'. 



Formulae for E, J, G, dx. in terms of k, § 2. 



2K 2E 



§ 2. The following formulae give the expansions of , — , &c. 



7T 7T 



in ascending powers of Jc 2 : 



2K _* l 2 , 2 1 2 .3 2 l a .3 2 .5 2 76 



w - l + 2 s + S^ 2 ^ 2 + 2 2 . 4 2 . 6 2 + ' 



?* -1 1 i* V - 3 y 1 '- ff - 8 y^ 



it 2 ! 2\4 a 2". 4*. 6* ' 



- = g V+WT* k +FT4V6 F + &a ' 



2G 1 79 l 2 74 l 2 • 3 2 7R 



V = 2 * +2\4 * 4 + f T T77 6^ + &c., 



18 »+£*,+*, 



7T 2.4 2 2 .4.6 



2(^+g) _ 1 . i „ , i 2 „ . r.3 2 



it ~ i+ 2 2fC + 2 2 .4 2 * + 2\4\6 



2(E-J) _ 1.3 1 2 .3.5 1 2 .3 2 .5.7 

 7T " _1 2 2 2 2 .4 2 2 2 .4 2 .6 2 



2T 2C 2T\ 



The formulae for — and — may be deduced from those for — and 



It IT IT 



2E 



— , either by means of the algebraical formulae 



7T 



J=K-E, G = E-(l-k 2 )K, 

 or by means of the differential formulae 



J = — k ~yt > G = k ( 1 — k) — jy" • 

 vol. v. pt. ill. 13 



