1885.] K', E, J', G' in ascending powers of the modulus. 249 



Change of q into iq\ § 48. 



§ 48. The change of q into iq* is equivalent to the change of 

 q into 5* followed by the change of q into — q. As shown in § 23 

 (p. 203), the change of q into iq? converts k' into e~ 2ie and K' into 

 ±e' e (K' — iK), where 6 denotes the modular angle, given by the 

 equation h = sin 6. 



Transforming the results in the last column of Table III. by 

 changing q into — q we find that, by the change of q into 17* 



E' becomes e~ i0 [Gf + WK+i (G + kk'K')}, 

 T „ e- ie (W' + iW), 



G' „ e~ ie [G' - kk'K+ i{G - klc'K')}, 



IT „ \e~* [G' + W - kk'K + i (G + W- kk'K% 

 V „ e- ie (G' + iG), 



W „ ±e- id {G'+W' + kk'K+t(G+ W+kk'K% 

 Since, by this change 



k' becomes e~ 2ie , 

 and K' „ y e (K'-iK\ 



it follows that 



k'K 1 



\e~ i6 {K' -iK). 



The transformations of K', T, V were given in § 24 (p. 203), 

 and by their means the series for 



K IC 2W 2W 4£ 4c7 

 > > j > ) ) 



7T 7T 7T 7T 7T 7T 



in terms of sines and cosines of multiples of the modular angle 

 were found in § 25 (p. 204). 



The transformations of E', G', V, W give no fresh results, 

 except the series 



4>kk'K 



IT 



4>kk'K' 



13 I 2 7 l 2 3 2 11 



= cos0 — 7^- cos 30 — ^—r. , cos 70 — ^ ,,' ^ cos 11<9 — &c, 



2 2 . 4 2 



2 2 . 4 2 . 6 

 1 2 .3M1 



13 l 2 7 



^ = sin + -^- sin 30 + ^ sin 7e + 2 *. 



77 sin 110 + &c. 



