ON THE RECENT PROGRESS OF ANALYSIS. 305 



dependence, the existence of which we are obliged to deny, may- 

 be expressed by a differential equation of the second order ; and 

 therefore it would seem that the commonly received opinion that 

 every differential equation of two variables has a primitive, or 

 expresses a functional relation between its variables, must be 

 abandoned, unless some other mode of escaping from the difficulty 

 is discovered. It is probable that some simple consideration, 

 rather of a metaphysical than an analytical character, may here- 

 after enable us to form a consistent and satisfactory view of the 

 question, and this I believe^ I may say is the opinion of M. 

 Jacobi himself. The same difficulty meets us in all the Abelian 

 integrals : as in the case of those of Legendre's first class, namely 

 where X is of the fifth or sixth degree, so also generally, the in- 

 verse function has more than its due number of periodicities. 



Abel, in a short paper in the second volume of his works, 

 p. 51, has in effect proved the multiple periodicity of the func- 

 tions which are inverse to the integrals to which his theorem 

 relates. The difficulty to which this gives rise did not strike 

 him, or was perhaps reserved for another occasion. 



M. Jacobi next proves that his inverse functions of two 

 variables are quadruply periodic, but that quadruple periodicity 

 for functions of two variables is nowise inadmissible. 



A difficulty however seems to present itself, which is sug- 

 gested by M. Eisenstein in Crelle's Journal, viz. that if for each 



/* dx 

 value of the amplitude the integral <f>x or I -= (vide supra, p. 



-' Vj" 



302) , has an infinity of magnitudes real and imaginary, and the 

 same is the case for (f>y, it is by no means easy to attach a defi- 

 nite sense to the equation u = <frx + <py, or to see how the value 

 of u is determined by it*. 



31. Two divisions of the theory of the higher transcendents 

 here suggest themselves, which are apparently less intimately 

 connected than the corresponding divisions in the theory of 



* The difficulty here mentioned may perhaps be met by saying that the value of 



(j>x determined by the integral I - is necessarily determinate, and so likewise 



Jo i^jX 



is that of u. That considerations connected with the conception of a function 

 inverse to <j>x make the latter quantity appear indeterminate is undoubtedly a 

 difficulty ; but it is, so to speak, a difficulty collateral to M. Jacobi's theory,_and 

 therefore need not prevent our accepting it. 



20 



