ON THE RECENT PROGRESS OF ANALYSIS. 313 



that of the integrals of any algebraical function whatever. Thus 

 his views bear the same relation to Abel's general theory, deve- 

 loped in the Savans Etrangers, that those of M. Jacobi in the 

 Considerationes Generates do to Abel's theorem. 



All that has yet been published with respect to them is con- 

 tained in the Comptes Rendus, xviil. p. 1133, in the form of an 

 extract of a letter from M. Hermite to M. Liouville. This ex- 

 tract is reprinted in Liouville's Journal, IX. p. 353. It was com- 

 municated to the Institute in June 1844. 



Following the course of^M. Jacobi's inquiries, M. Hermite 

 proposed to determine what are the differential equations of 

 which Abel's investigations give the complete algebraical inte- 

 grals. When this is done it suggests the nature of the inverse 

 functions which are to be introduced. The number of these 

 functions will of course vary in different cases, just as in M. Ja- 

 cobi's less general theory. Let us suppose this number to be 

 denoted by 7, then each function will involve 7 variables. And 

 if each of these variables be replaced by the sum of two new 

 variables, then all the functions are given as the roots of an 

 equation of the 7th degree, whose coefficients are rational in 

 terms of the corresponding functions of each of the new variables 

 and of certain known algebraical functions. From hence is de- 

 rived the theory of the periodicity of these functions. 



After some other remarks on the theory of the higher trans- 

 cendents, M. Hermite states that the method of division of which 

 he made use in the problem of the division of Abelian integrals 

 extends also to the new transcendents now considered, but that 

 in the theory of transformation he had not as yet been success- 

 ful. The greater part of the remainder of this remarkable com- 

 munication relates to elliptic functions, and has been already 

 noticed. The remark just mentioned as having been made by 

 M. Jacobi for the functions which are inverse to the Abelian 

 integrals, extends, M. Hermite observes, to the functions which 

 he considers. 



In conclusion, M. Hermite remarks that the method of dif- 

 ferentiation with respect to the modulus of which Legendre made 

 so much use in the theory of elliptic functions, may be applied 

 to all functions of the form 



jf(xy] dx, 



