248 ON THE RECENT PROGRESS OF ANALYSIS 



instance, if J/and N are general symbols denoting any integral 



Vlf -, M 



functions of x, the two suppositions y = ^- ancl y = "r^ ar( 



precisely equivalent, since by an obvious reduction, and by 

 changing the signification of M and N t the one may be trans- 

 formed into the other ; and so in more general cases. Thus the 

 same function may assume a variety of aspects, and there will 

 be a corresponding variety in the form of our final results. 



In Crelle's Journal we find a good many essays on this part 

 of the subject : of these I shall now mention several. 



M. Broch is the author of a paper in the twentieth volume 

 of Crelle's Journal, p. 178. It relates to the integration of certain 

 functions irrational in consequence of involving a polynomial of 

 any degree raised to a fractional power. For these functions he 

 establishes formulae of summation, which of course include Abel's 

 theorem, since the latter relates to cases in which the fractional 

 power in question is the (J)th. Subsequently to the publication 

 of this paper he presented to Jthe Institute a memoir on the 

 same subject, but gave to the functions to be integrated a 

 different but not essentially more general form. This memoir, 

 which was ordered to be printed among the Savans Etrangers, 

 but which will be found in Crelle's Journal (xxin. 145), may 

 be divided into two portions : the first contains results analogous 

 to Abel's theorem: the second relates to the discussion and 

 reduction of the transcendents which they involve. In this part 

 of his researches M. Broch has followed the method, and oc- 

 casionally almost adopted the phraseology of a memoir of Abel, 

 on the reduction and classification of Elliptic Integrals (Abel's 

 Works, II. p. 93). MM. Liouville and Cauchy, in reporting on 

 the memoir, conclude by remarking that the author *n'a pas 

 trop presume* de ses forces en se proposant de marcher sur les 

 traces d'Abel.' 



M. Jiirgenson has contributed two papers to Crelle's Journal 

 on the subject of which we are speaking. The first, which 

 is very short, contains a general theorem for the summation 

 of algebraical integrals* when the function to be integrated is 

 expressed in a particular form. This paper appears in the 



* I have used the expression " algebraical integrals," though perhaps not cor- 

 rectly, to denote the integrals of algebraical functions. 



