1880.] Inverse Elliptic Functions. 369 



and generally (p. 82) the integral / ti dx, where X denotes a 



sextic function, whose skew-invariant vanishes, can be similarly 

 expressed. 



For if the skew-invariant vanishes, the roots of the sextic 

 form an involution, and by a linear substitution we can make 

 the sextic a reciprocal expression. 



We have 



^argsn(^^i^^,, tan Itt 





= — Yn — ^ — ^f iT^ > V2 — 1 > tan ^tt; 



and _argsn\^-X___, tan-i^^ 



_ 1 + (V2 - 1) x' 



Therefore, x < \/tan Itt, 



1-a;' 



p d^ V2-1 /1~^' V2 + 1 "'Vl + ^' 



/ = o /^ argsna;. / -- — — , + ,-, ,^ arg sn 



JoJ\-x' 2V2 ^ V l-^--^ ^' 



1-h^' 2V2 ^ tan^TT 



p a;'cZ^ 1 /l-^'^ 1 



I -r^i R^ = ~ -E^—^ arg sn ^r * / z ^ + ^7-777 arg sn 



JoV(l-«) V^ Vl + « V2 ^ taniTT ' 



and X > J tan ^ir, 



/ 1-x 



fi dx V2-1 /1-x' J2 + 1 ""yi-vx 



—ii^ 8\ = — s—To- a-rg sn a? . / 5 4- -V-7?^ arg sn — i-= — -— 



— x 



^ ic'^cZa:; 1 /I — ic^ 1 'V 1 -f- X" 



argsn^./— ?• + ;t— 7;^ arg sn 



ll-i 



.,^il-x') 2 V2 *= V 1 + x' 2 V2 ^ tan Itt 



(Richelot, Cre^fe, t. 32, p. 213). 



27-2 



