82 . REPORT—1846. 
At the close of his memoir, M. Richelot has given some numerical exam- 
ples 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 com- 
parison of transcendents was studied principally by German and Scandi- 
navian writers*: a young French mathematician, M. Hermite, has recently 
made important discoveries in this theory ; but as the principal part of what 
he has done is as yet not published, a very imperfect outline is all that can 
be given. 
In the seventeenth volume of the ‘Comptes Rendus’ we find the report 
of a commission, consisting of MM. Lamé and Liouville, on a memoir pre- 
sented 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 divi- 
sion of elliptic integrals, led to a very warm discussion between MM. Liou- 
ville 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 asuming, as in the analogous 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 transcen- 
dental 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 justify 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 gene« 
ralisation. 
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 important discovery he had made, he points out 
that the transcendental functions A (wv), A, (wv) (vide ante, p.'76) are alge- 
braical 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, that of the integrals of 
any algebraical function whatever. ‘Thus his views bear the same relation 
to Abel’s general theory, developed in the ‘ Savans Etrangers,’ that those of 
M. Jacobi in the ‘ Considerationes Generales’ do to Abel’s theorem. 
All that has yet been published with respect to them is contained in the 
“Comptes Rendus,’ xviii. p. 1133, in the form of an extract of a letter from 
M. Hermite to M. Liouville. This extract is reprinted in Liouville’s Journal, 
ix. p. 353. It was communicated 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 integrals. When this is done it suggests the 
* The papers of M. Liouville, already noticed, may be said to be an exception to this 
remark, 
