1885.] K', E', J', G' in ascending poivers of the modulus. 235 



if we substitute for the quantities involved the values assigned to 

 them by the equations : 



E = E, E'=l'+K', 



i = e-k, r = r, 



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



U = E-} 2 {\ + k' 2 ) K, U' = I' + WK\ 



V = E-\ k' 2 K, V = T + \ (1 + k") K', 



W = E-\K, W = /' + \K'. 



Thus, for example, equation (xx) becomes 



E{F + |(1 + k' 2 ) K'\-I' [E-l{l 4 k' 2 )} K= J(l 4 k' 2 ) it, 



that is 



i (1 4 k' 2 ) {EK! 4 I'K) = 1(1 4 k' 2 )7r, 



which is equivalent to (i). 



The systems of formulae which express E, I, G, U, V, W in 

 terms of any one of them and K are given in § 45. 



§ 34. It will be noticed that in the 21 formulae (i)...(xxi) 

 each letter occurs in combination with every other letter except 

 the one to which it corresponds. Taking, for example, E we find 

 the combinations 



EK', EE', EG', EU', EV, EW, 

 but not EI'. Similarly we find 



G K', GE', GT, GU', GV, GW\ 

 but not GG '. 



Relations involving K x , K', E t , E', <&c, § 35. 



§ 35. Using K v E v I v &c. as in § 9 (p. 191), to denote 



2K 2E 27 



— , — , — , &c, the 21 equations may be written : 



7T 7T 7T 



(i) E X K' +I'K t =1, 



(ii) I t K' +E'K 1 =1, 



(iii) G t K' + G'K 1 =l, 



(iv) U X K' 4 TK X = \, 



(v) V t K' 4 U'K 1 = 1, 



(vi) W X K +WK X = \, 



(vii) EJL' -I J' =1, 



