294 ON THE RECENT PROGRESS OF ANALYSIS. 



After establishing these results in the memoir (that on ellip- 

 tic transcendents of the first and second kinds), which will be 

 found in the twenty-third cahier of the Journal de TEcole 



Polytechnique, p. 37, M. Liouville supposes y to be of the 



p 

 form -jjj , where P and R are integral polynomials, and hence 



f P 



deduces the general form in which the integral \-pp dx may 



J Y i 



necessarily be put, provided it admit of expression as an ex- 

 plicit finite function of x. 



f P 

 He shows from hence that if I p^ dx cannot be expressed 



by an algebraical function of x, it cannot be expressed by an 

 explicit finite function of it, and finally demonstrates that an 

 elliptic integral, either of the first or second kind, is not ex- 

 pressible 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 VEcole Poly- 

 technique, 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 dis- 

 tinctions between different classes of functions, states that he 



had succeeded in showing that the integral I 



JV 



- s 



not expressible as a finite function, explicit or implicit, of x. 

 Laplace however did not publish his demonstration. 



In his own Journal (v. 34 and 441), M. Liouville has since 

 shown that elliptic integrals of the first and second kinds, con- 

 sidered as functions of the modulus, cannot be expressed in finite 

 terms. 



25. In the eighteenth volume of the Comptes Rendus 

 (Liouville's Journal, ix. 353), we find in a communication from 

 M. Hermite, of which we shall shortly have occasion to speak 

 more fully, a remarkable demonstration of Jacobi's theorem. It 

 is stated for the case of the first real transformation, but might 

 of course be rendered general. This demonstration depends 



