1880.] 



Inverse Elliptic Functions. 



367 



/"/ 7S-3 7 // 2 \ (a-'h — 1\jx—a.x—h 1 



arg en 



a— cc. x—h) '^ dx = 





V(a - ^') 



arcf en 



3 2 



{(a;-m)^+?2,^] 4c^^=_^ argcn 



((a — x.x—hy 1 ] 



j(^_,,^)2 + ^2||, ^2j- 



r2 * 



(1 - X")-^ dx {T^ > £C> 0) 



•J X 



= r_ (1 - x')-^ dx{l>x>.2-^) 



(23) 



c?^ 



(ic— a. iT — 5. cc — c)' 



{4< .a — b . a — c .b — cY 



b-G.x-ay (\/S — l) — (4!. x—h .x—c. a— b. a — cy 



__ - 



{b—c.x—ay{i\/^ + V)-{-{'4.x—h.x-c.a—b.a—cY 

 dx 



arsf en 



, sin75'' 



(24) 



[[x — a) {{x — my + n""]]^ 



3« 



2n^ {{a - my + 7f]^ 

 argen 



n.x- ayH^/S + l)-{{x-my+n\{a-my+n'']^ 



-,, sin 15° 



_{n .x-ay (V3-1) + [{x-my+ w^ {a - my+7i^]^ 



an integral oecurring in the motion of a body moving under gravity 

 in a medium in which the resistance varies as the cube of the 

 velocity. The last two integrals (23) and (24) were considered 

 by Allegret {Comptes Rendus, t. QQ, p. 1144). 



If X = l — X .1 + Kx .\ + \x .1 — Kkx (Cayley, Elliptic Func- 

 tions, Chap. XVI.), 



VOL. III. PT. VIII. 27 



