1884.] K', E', J', G' in powers of the modulus. 207 



Gudermann s form of the series for K' and E', § 29. 



§ 29. In Vol. xix. pp. 55 — 58 of Grelle's Journal, Gudermann 

 gave the series for K' and E' in Legendre's form and also in the 

 following slightly different form : 



Let 



12 -1 2 02 -1 2 02 r2 



= 1 + 2-2^+^742^ + 2 / 4 »' 6 « y + &C - 



and 



(J -i+^ + l^V . l'.3'.5...(2n-l)' 

 Jt »-- L ^2 2 2 2 .4 2 2 2 .4 2 .6 2 ...(2w) 2 ' 



(so that £ n denotes the first n + 1 terms of i?), then 

 iT = 5Uo 

 Similarly, let 



^ = ^log|-(^-l)-3^ i (^-^ 1 )-^.(i£-^)-&c. 



and 



2 + F74 + 2\4*.6 ' 



Zn 2 K + VA - + a- 2*.4 a ...(2 M -2) s 2w ' 



(so that £ B denotes the first w terms of t, the last term being 

 halved), then 



E'=t\og* c + I-(t-t 1 )-^(t-t 2 )-^- Q (t-Q-&c. 



It does not seem worth while to give the corresponding forms 

 of the series for I', G' t &c; they may be derived at once from the 

 formulas in § 3. 



Weierstrass's J and J', § 30. 



§ 30. In Weierstrass's notation, in which K — E is denoted 

 by / and E' by J' (see § 7), we have, as noticed by Weierstrass 

 himself*, the corresponding approximate formulae : 



er = i + ^io g |. • 



IT A, 



* Crellc's Journal, Vol. lii. p. 364. 



