ON THE RECENT PROGRESS OF ANALYSIS. 311 



p. 221. The aim of it is to furnish the means of actually calcu- 

 lating the value of the Abelian integral of the first class by a 

 method of successive transformation, that is, by a method analo- 

 gous to that used for elliptic integrals. M. Eichelot's process 

 depends essentially on an irrational substitution, by means of 

 which we can replace the proposed integrals by two others which 

 differ only with respect to their limits. In the development of this 

 idea the author confines himself to the first kind of the Abelian 

 integrals of the first class, though the same method may m . m . 

 be more generally applied*. ^From the formula which expresses 

 the proposed integral as the aggregate of two others is deduced 

 another, in which it is expressed by means of four integrals, 

 the inferior limits of all being zero. The first and second of 

 these integrals differ only in their amplitude, and the same is 

 true of the third and fourth. There are two principal trans- 

 formations, either of which may be repeated as often as we 

 please ; and though it might seem that the number of inte- 

 grals would in the successive transformations increase in a geo- 

 metrical progression, yet by the application of Abel's theorem 

 we can always reduce them to the same number. But the de- 

 velopment of this part of the subject M. Eichelot has reserved 

 for another occasion f. 



At the close of his memoir, M. Eichelot has given some 

 numerical examples of his method for the case of a complete 

 hyper-elliptic integral. The third example he had previously 

 given in a brief notice of his researches, published in No. 311 of 

 Schumacher's Journal. 



33. For many years after the death of Legendre the subject 

 of the comparison of transcendents was studied principally by 

 German and Scandinavian writers \: a young French mathema- 

 tician, M. Hermite, has recently made important discoveries 

 in this theory ; but as the principal part of what he has done 



* The integral to be transformed is 77 ; - r - 



- r - ^ . 9 \ 

 (1-2) (i -/c%) (i -X 2 z) (i - 



M and N, &c. being constant. 



"t* His transformations ultimately reduce the hyper- elliptic integral to elliptic 

 integrals j the latter may be considered known quantities, "vel per paucas ad- 

 jectas transformationes directe computentur." 



+ The papers of M. Liouville, already noticed, may be said to be an exception 

 to this remark. 



