312 ON THE RECENT PROGRESS OF ANALYSIS. 



is as yet not published, a very imperfect outline is all that 

 can be given. 



In the seventeenth volume of the Comptes Eendus, we find 

 the report of a commission, consisting of MM. Lame and Liou- 

 ville, on a memoir presented to the Institute by M. Hermite. 

 This report is reprinted in the eighth volume of Liouville's 

 Journal, p. 502. A remark which incidentally occurs in it, 

 namely, that Abel was the first to give the general theory of the 

 division of elliptic integrals, led to a very warm discussion be- 

 tween MM. Liouville and Libri, on the subject of the claims 

 which, as I have already remarked, the latter had made with 

 reference to this theory. 



It appears from the report, that M. Hermite has succeeded 

 in solving the problem of the division of hyper-elliptic integrals. 

 The division of elliptic integrals depends on the solution of an 

 algebraical equation ; that of the hyper-elliptic integrals (as the 

 functions inverse to them involve, as we have seen, more than 

 one variable), on the solution of a system of simultaneous 

 algebraical equations. This solution can, M. Hermite has 

 shown, be effected by means of radicals assuming, as in the ana- 

 logous case of elliptic functions, the division of the complete 

 integrals. M. Hermite's method depends for the most part on 

 the periodicity of the functions considered. A transcendental 

 expression of the roots of the equation of the problem having 

 been obtained, their algebraical values are deduced from it. 



These researches, in themselves of great interest, are yet 

 more interesting, when we consider how completely they jus- 

 tify the views of M. Jacobi as to the manner in which Abel's 

 theorem ought to be interpreted, by showing that his theory 

 of the higher transcendents is no barren or artificial generali- 

 sation. 



At page 505 of the volume of Liouville's Journal already 

 mentioned, we find an extract from a letter of M. Jacobi to 

 M. Hermite, in which, after congratulating him on the im- 

 portant discovery he had made, he points out that the transcen- 

 dental functions X (uv), \ (uv) (vide ante, p. 302) are algebraical 

 functions of transcendental functions which involve but one 

 variable. 



M. Hermite's subsequent researches have embraced a much 

 more general theory than that of the Abelian integrals, namely, 



