278 ON THE RECENT PROGRESS OF ANALYSIS. 



case in question is soluble by a method closely analogous to that 

 used by M. Gauss for the solution of binomial equations. Thus 

 for all such values the problem of the division of elliptic inte- 

 grals is completely solved. 



The most remarkable of these cases corresponds to the 

 geometrical problem of the division of the perimeter of the 

 lemniscate. Abel discovered that this division can always be 

 effected by means of radicals, and further, that it can be con- 

 structed by the rule and compass in the same cases (that is for 

 the same values of the divisor) as the division of the circum- 

 ference of a circle. Of this discovery we find Abel writing to 

 M. Holmboe, " Ah qu'il est magnifique! tu verras*." 



In order to form an idea of the nature of the difficulty which 

 disappears in the case of which we are speaking, let us suppose 

 that we have to solve the algebraical equation which is repre- 

 sented by the transcendental one < (30) = 0, in the same manner 

 as the equation 4# 3 3x = is represented by sin (36) = 0. 



The roots of 4o? s 3x = 0, are, setting aside zero, 



. 2?r . 4?r 

 sm T , sin . 



Those of the former algebraical equation, which, as we know, is 

 of the ninth degree, are, beside zero, 



2&) 4o) 



2m* 



~T' T 



2 (w 4- m") 4 (to + m') 



+ 2m') , 4 (a) + 

 -- ' * -- 



where * = V 1. 



* It is right to mention that M. Libri has disputed Abel's title to the theory 

 of the division of the lemniscate. I shall, however, not enter on the merits of the 

 controversy which arose on this point between him and M. Liouville. The reader 

 will find it in the seventeenth volume of the Comptes Rendus. It appears that 

 M. Gauss had himself recognised the applicability of his method to the equation 

 arising out of the problem of the division of the perimeter of the lemniscate (vide 

 Eecherches Arithmetiques, vii. p. 419. I quote from the translation published 

 at Paris, in 1809). 



