Applications of AbePs Theorem, 1 1 7 



glish language, to my knowledge, the following examples of 

 its application to some of the simplest cases which can be pro- 

 posed are here offered, with a view of attracting attention to 

 an important analytical discovery*. Legendre has applied 

 Abel's theorem to the integral 



dx 



/-. 



The subject of the first example here detailed is the integral 



dx 



f- 



4/1-^ 



which may, in fact, be reduced to an elliptic integral of the 

 first kind ; and thus, through the well-known integral due to 

 Euler, a confirmation may be easily obtained, in this instance, 

 of the result found by the method of Abel. 

 The subject of the second example is 



dx 



f 



which cannot in general be reduced to an elliptic integral. 

 Here I have chosen for the equations of condition between 

 the limits x^^ jr^, x^;^ &c., equations similar to those employed 

 by Mr. Talbot in his paper on the sum of three arcs of the 

 equilateral hyperbolaf. I have also applied the method of 

 Abel to the integral 



x^dx 



f 



Vl+o^*' 



and deduced therefrom the theorem given by Mr. Talbot in 

 the paper to which I have referred. 



The theorem of Abel is as follows : 



" Soit <^x une fonction entiere de .r, decomposee d'une 

 maniere quelconque en deux facteurs entiers ^jj x et ^^ x^ en- 

 sorte que ^ x = ^■^x,<^^x» ^oilfx une autre Ibnction entiere 

 quelconque et 



**-y(*-«)v/(<^"^)' 



ou a est une quantite constante quelconque. Designons par 



Aq, «!, %, .... ; Cq, q, Cg, des quantites quelconques dont 



Tune au moins soit variable. Cela pose, si Ton fait 



* Professor Jacobi says, ** Wir halten es, wie es in einfacher Gestalt 

 ohne Apparat von Calcul den tiefsten unci umfassendsten mathematischen 

 Gedanken ausspricht, fUr die grdsste mathematische Entdeckiing unsrer 

 Zeit, obgleich erst eine kiinftige, vielleicht spate, grosse Arbeit ilire ganze 

 Bedeutung aufweisen kann. — Crelle's Journal, vol. viii. p. 415. 



t [ A notice of Mr. Talbot's paper will be found in Lond. and Edinb. 

 Phil. Mag. vol. iv. p. 225.— Edit.] 



