1880.] Inverse Elliptic Functions. 365 



dx 



(19) 



6 \/[- (^ - a) (^ - ^) {(«; - nif + n'W 



1 (/3 (a; -a) + a{x-h) J 



- argcn ; J J-, k\ , 



V(ay8) ^ l/3(a;-a)-a(a;-&) 

 dx 



rx 



(20) 



s/[{x-a)[{x-7rif + n'W 



1 fa — (^ — a) , I 



-r arg en ^ — r^ ( , A;^ , 



Va (a + cc - a) ' ' ' 



where (a — mf + n" = a^, 



^^^^ i V[-(^-f^){(^-^)' + w'}] 



1 /x — a + a , \ 



= -7^ arg en , k] , 



J a \x-a-a J 



where ^^ = A I 1 



(22) 



2 



dx 



V{(^ - 771)'' + rt'''} {(a; - mf + n"] 

 is reduced by putting 



x — m = n tan [ji — co), 

 and then = 



d(f) 



A/[{(m-my+?i'+?i"'}+|{(m-m')''— ?^^+?^''}cos2((/)-(»)-?i(m-??^')sin2(0-&))]' 



The coeffieient of sin 2<p is made to vanish by putting 



_ 2n (m — m') 



tan 2ft) = \^ 5^ — jr, , 



[m — iii) —n +n 



and then, if 



r'^ = [m ~ m'f + {n + n-')^ 



s^ = (m — m'Y -\-(n — n'Y ; 



