1898.] differential equations defining periodic functions. 515 



express the partial differential coefficients of the third order in 

 terms of x, y, z, %, 77, £, in the forms 



d 3 w 



9^i = ^ etc., 



where P and /a 2 are rational in x, y, z, f, rj, £ We obtain several 

 different forms for /jr in a similar way. It is supposed that the 

 differential equations are such that these are algebraically 

 equivalent in virtue of the equations f = 0, f 2 = 0, /, = 0. 



Now we have 



3 3 ?i> (Pw c) s w 



d,r= dn> + =- j du« + _ „ 7 dvn = a (Pdu 3 + Qdu 2 + Rdvn), 



dii.;' dun-du, 2 0n.rd,u^ 



and similarly, 



dy = fi (Qdit, + P^dn, + Qidiij), dz = fi (Rdu, + Qidu., + R^lu^) ; 

 thus we have 





efo; + i/ffy/ + Gdz), 



u, = f- (Hdx + Bdy + Fdz), 

 J ft 



*h = - (Gdx + Fdy + Gdz), 

 J & 



where A, B, G, F, G, H. ft? are rational functions of x, y, z, £, v, £ 

 these latter being connected by the rational equations f l = 0, 

 / a = 0,/ 3 = 0. 



Therefore, if the original differential equations have a 

 solution of which the second partial differential coefficients are 

 single-valued functions, these integrals will be capable of inversion, 

 and define x, y, z, £, rj, £ as single-valued, in general simultaneously 

 periodic, functions of u 3 , u 2 , v^. Whence the solution of the 

 differential equations can be completed. 



In what follows we obtain systems of differential equations of 

 the form just considered, the system with p independent variables 

 being satisfied by all hyperelliptic theta functions of p variables, 

 independently of the characteristic of the theta function (the 

 dependent variable w being the negative logarithm of the theta 

 function), and we verify in detail for the cases of two and three 

 variables that the necessary conditions for the application of the 

 method of integration sketched above are satisfied. The method 

 would appear to be of importance as an elementary introduction 

 to the theory of theta functions, but it has a wider application, 

 namely to the general theory of functions with simultaneous 

 period-systems. 



41—2 



