1880.] Inverse Elliptic Functions. 871 



Therefore 1 - x\ x' - k'' 



= (1 + n^) x^ -x^- n^ 

 = (1+ n') x^ - {x' + ny + 8a;V {x" + n^) 

 = {l+n')x^-x'z^^^x%\ 

 '1 x~^dx 



and Z— % 



\/(l + n^ + 3w'^ - /) 



+ - 



\/(^r + 272) /^{z—2n) 



z V(i + ^^^ - z) [[^ + HI + ^^'Tl + f (1 - '^"fl 



4(l+w + w')i(l-^ + ?i')^ 



[ (^ + 272) V3 (1 + 92 + w^)^ - (1 + ^' - ^) (1 - ^ + ^')^ ^ 

 [(^ + 2w) V3 (1 + ?i + n^^ + (1 + ?2' - ^) (1 - w + 71^^ ' ' 



arff en 



+ 



3? 



4(l + n + 72')^(l-« + w')^ 



[ {z + 2n) V8 (1 - 72 + n^- - (1 + ^' - g) (1 + n + ?i')^ ^ , 

 ^'^''''V + 2n)V3(l-72 + /2'0^+(l+»i'-^)(l + w + w')^' ''"' 



where 



yt^=lU-^ (l-72 + r2T-3..^ 



2 (1 + n + W'}^ (1 - 92 + w')^'] ' 

 by (14). 



..^^ f, V3 (1+72 + 72^-322^ 



' M 2 (1 + ^ + ,,^)! (1-^ + ^^)^ 



Therefore 



,sj{^-x\x'-'y'') 



2 -^3 (1 + A;'* + A;'^)^ (1 - A;'^ + A;'^)^ 





+ 



2 v'3 (1 + A;'^ + ^"'*)* (1 - Ic^ + A;'t)i 



(^+l^'^)V3 {l-k'^+0y^-{l-x ){x-0){l+0+Jc'i)h ^ 



arof en 



(a;+ A;'^)V3 (1 - A;'^ + ^^'¥ + (1 -^) (^-A;'^) (1 +A;'^+A;'^)i 



