1886.] inverse to the second elliptic integral, 373 



It is convenient to denote am ea u by ame u and to use sne u 

 to denote sin ame u = sn ea u = sin am ea u, cne to denote cos ame u, 

 &c. Denoting E — K' by /', the function ea x is such that 



ea (x + 2E) = ea x + 2K, 



ea (x + 2il') = ea,x— 2iK', 



ea (x + 2E + 2iF) = ea x + 2K - 2UC. 



Taking the am of these equations we find 



ame (x + 2E) = ame x + ir, 



ame (x + 2il') — — ame x + tr, 



ame (x + 2E + 2il') = — ame x. 



The function amex is therefore periodic with respect to 4>E+4<iI' 

 and quasi-periodic with respect to 4<E or 4<il'. 



Taking the sn, en, dn of these equations we find that sne x, 

 cne x, dne x are doubly periodic, the periods being 4<E and 4il'. 



Corresponding to the formulae in Elliptic Functions in which 

 the argument is increased by a quarter-period we have the equa- 



tions 



/ -rr 7 o sne x cne x\ Tr 



ea ( x + E — k — j J = ea,x + K, 



( . T , sne#dne#\ , Tr . 



ea [x + il = ea x - iK , 



\ cne x ) 



( „ . T , cne x dne x\ rr 



ea [x 4- E + il ' + — = ea x + K - iK , 



V sne x J 



so that, for example, corresponding to 



x + K ) = 1 , 



dn x 



we have 



/ 7-, 7 2 sne x cne x\ cne # 

 sne # + Z? — k 



dne« / dne a;" 



In general, in results in Elliptic Functions in which the argu- 

 ments are u, v, u + v we may replace sn, en, dn's by sne, cne, due's 

 if we replace the argument u + v by u + v — E u% v (leaving the argu- 

 ments u, v unaltered), where E Ut v is a certain function of the sne, 

 cne, dne's of u and v. 



The author had considered not only the functions ea x, ame x, 

 sne x, &c. but also the complete system of functions ia x, ga x, ea x, 

 ami #, amg#, ame#, sni x, sng#, sne a;, &c. obtained by inverting 



