70 REPORT—1846. 
same subject we may mention a paper by M. Eisenstein (Crelle’s Journal, 
Xxvii. 285). 
24. M. Liouville has in several memoirs investigated the conditions under 
which the integral of an algebraical function can be expressed in an alge- 
braical, or, more generally, in a finite form. This investigation is of the 
same character as that which occurs in the beginning of Abel’s last published 
memoir on elliptic functions (vide supra, p. 65). But while Abel’s re- 
searches are more general than M. Liouville’s, the latter has arrived at a 
result more fundamental, if such an expression may be used, than any of 
which Abel has left a demonstration. 
He has shown that if y be an algebraical function of 2, such that J yd 
may be expressed as an explicit finite function of x, we must have 
Sydaatt Alogu + Blogv +...+ Clog w, 
A, B,...C being constant, and ¢, uw, v,...w algebraical functions of a. 
The theorem established by Abel in the memoir referred to includes as a 
particular case the following proposition, that if 
Syde =t+ Alogu+ Blogy+...+ Clogw, 
then ¢, u, v, ...w may all be reduced to rational functions of # and y. 
Combining these two results, it appears that if #2 yd« be expressible as an 
explicit finite function of x, its expression must be of the form 
t+ Alogu + Blogv +...+ Clogw, 
where ¢, w, v,... w are rational functions of x and y, or rather that its ex- — 
pression must be reducible to this form*. 
After establishing these results in the memoir (that on elliptic transcen- 
dents of the first and second kinds), which will be found in the twenty-third 
cahier of the ‘ Journal de l’Ecole Polytechnique,’ p. 37, M. Liouville sup- 
poses y to be of the form , where P and R are integral polynomials, and 
hence deduces the general form in which the integral cd xz may neces- 
sarily be put, provided it admit of expression as an explicit finite function of x. 
rE 
VR 
braical function of xz, it cannot be expressed by any explicit finite function 
of it, and finally demonstrates that an elliptic integral, either of the first or 
second kind, is not expressible as an explicit finite function of its variable. 
In a previous memoir inserted in the preceding cahier, M. Liouville 
proved the simpler proposition, that elliptic integrals of the first and second 
kinds are not expressible as explicit algebraical functions of their variable 
(Journal de l’Ecole Polytechnique, t. xiv. p. 137). His attention appears to 
have been directed to this class of researches by a passage of Laplace’s 
‘ Theory of Probabilities,’ in which the illustrious author, after indicating the 
fundamental, and, so to speak, ineffaceable distinctions between different 
classes of functions, states that he had succeeded in showing that the inte- 
He shows from hence that if dz cannot be expressed by an alge- 
d 
geal [ae is not expressible as a finite function, explicit or 
implicit, of z. Laplace however did not publish his demonstration. 
* An equivalent theorem is stated by Abel in his letter to Legendre for implicit as well 
as explicit functions (Crelle’s Journal, vi.). 
