ON THE RECENT PROGRESS OF ANALYSIS. 293 



therefore effected for Abel's developments that which M. Cauchy 

 had done for M. Jacobi's. Mr Cayley's paper appeared in the 

 fourth volume of the Cambridge Mathematical Journal, but he 

 has since published a translation of it with modifications in 

 the tenth volume of Liouville's. On the 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 algebraical, 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. 286). 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 x, such 



that \ydx may be expressed as an explicit finite function of x, 

 we must have 



\ydx = t + A log u + B log v + . . . + Clog w, 



A, B, ... C being constant, and t, u, v, ...w algebraical functions 

 of x. The theorem established by Abel in the memoir refer- 

 red to includes as a particular case the following proposition, 

 that if 



+ ... + Clogw, 



then t, u, v, . . . w may all be reduced to rational functions of 

 x and y. 



Combining these two results, it appears that if \ydx be ex- 

 pressible as an explicit finite function of x, its expression must 

 be of the form 



t + A logu + JBlog v+ ... 4- Clogw, 



where t, u, v, ... w are rational functions of x and y, or rather 

 that its expression must be reducible to this form*. 



* An equivalent theorem is stated by Abel in his letter to Legendre for im- 

 plicit as well as explicit functions (Crelle's Journal, vi.). 



