1880.] J/r Gr^eenhill, Integrals expressed by Elliptic Functions. 361 



(4) A. G. Geeenhill, M.A., Integrals expressed hy Inverse 

 Elliptic Functions. 



If I ~T7^ 2x = '^j then X =■ sin u, and u, the inverse function, 



Jo\/{l—x) 



is denoted by sin"' x by English writers, and by arc sin x by con- 

 tinental writers, 

 rx f^^ 



If I -~n~i — Tx = '^^> then x — h (e" + e ") = cosh 2t, and zf, the 

 Ji\/{x - 1)_ 



inverse function, is denoted by log {x + ^/{x'^ — 1)}, a notation uni- 

 versally employed. 



In ordinary treatises on the Integral Calculus it is shewn how 

 the integral of any rational expression involving /^R, where H is 

 of the first or second degree, leads to either of the above forms and 

 to algebraical expressions. 



When R is of the third or fourth degree, elliptic integrals are 

 introduced ; and from the definition of the function, 



•f [ ^ dx 



then ic = sn {u, k). 



To express u, the inverse function, in terms of x, we may either 

 use a notation employed by Gudermann, analogous to the conti- 

 nental arc sin, and put 



M==argsu {x, k), 

 or we may use the English notation and put 



u — sn~^ (x, k). 

 Then, from the definitions of the functions, (^'■^<1), 

 rx ^QQ 



^^^ j Jil-x' l-Fx^) ^ ^'^"^ ^^' ^^ °^ ^^'^" ^^ ^^' ^^' 



(2^ j. V(l-^-.A;- + A;V; ^ '"" ^^' ^^ '' "'^ '^ ^"' ^^' ■ 



C^ dx 



(3) j jn_^^^^^^_}.-^. = dn ' (^' ^■) o^ ^^S dn {^, f^)- 



The results of the substitutions to reduce elliptic integrals to 

 their normal form are now clearly exhibited when the elliptic 

 integrals are expressed by inverse elliptic functions, the form of 

 the result indicating the substitution to be employed. 



