ON THE REGENT PROGRESS OF ANALYSIS. 303 



Hence the functions X (u + u\ v + v l ) and \ (u + u 1 ,v+ v l ) are 

 expressible as algebraical functions of 



\(uv), \(uv), X(wV), \(wV). 



These then are functions to which the integrals are in a cer- 

 tain sense inverse, and which have the same fundamental pro- 

 perty as circular and elliptic functions. 



In the general case of Abel's theorem, we introduce (when 

 the degree of the polynomial is 2m or 2m 1), m 1 functions 

 analogous to X, each being a^ function of m 1 variables. These 

 functions will, it may easily be shown, have the fundamental 

 property just pointed out for the case in which m is equal 

 to three. 



Again, the differential equations of which Abel's theorem 

 gives us algebraical integrals, are, if the degree of the polynomial 

 X be five or six, the following : 



dx dy dz _ 



++ ' 



xdx ydy zdz 



+ + = 0; 



and generally, if the degree of the polynomial be 2m or 2m 1, 

 there are m - 1 such equations, the numerators of the last con- 

 taining the (m 2)th power of the variables. 



M. Jacobi concludes by suggesting as a problem the direct 

 integration of these differential equations, so as to obtain a proof 

 of Abel's theorem corresponding to that which Lagrange gave of 

 Euler's (vide ante, p. 241). 



30. Another important paper by M. Jacobi is that which 

 is entitled De Functionibus duarum Varidbilium quadrupliciter 

 periodicis, etc. (Crelle, xiil. p. 55). It is here shown that a 

 periodic function of one variable cannot have two distinct real 

 periods. In the case of a circular function, though we have for 

 all values of x 



sin x = sin (x + 2m?r) 

 = sin 



m and n being any integers, yet 2m7r and 2mr do not constitute 

 two distinct periods, since each is merely a multiple of 2?r, which 



