418 Dr. H. Bateman on some 



The function a satisfies the partial differential equation 



-dx-dz+ty-dy+dz -dz-f-dfdt' ' ' ' K } 



and j3 satisfies a similar equation. The functions X and Y 

 are to some extent indeterminate, since certain functions of 

 them will serve the purpose of § 1 just as well as X and Y. 

 "We shall suppose them to be chosen so that a depends only 

 on s and t, both of which satisfy (12). Choosing a so that 

 the condition 



is also satisfied, we find that there is a relation of the 

 form 



F(s, r)da = 2(c 2 -Z /2 - v ' 2 -? 2 )ds 



-.*-[{(r'+v)(«-r)+£"(e+v)} 



-.^{(r-vo^+o-rcr-V)}], 



where F(s, t) is a function whose value need not be specified 

 here. The condition a = const, or doi = leads to the Riccatian 

 equation which is used in finding the lines of electric force*, 

 and which has been solved by Dr. Murnaghan f in a large 

 number of cases. 



For our present purpose we need only the fact that 

 a depends only on 5 and t. An important consequence 

 of this is that u satisfies the three partial differential 

 equations of type 



d(«, t) __ i d(g, t) 



because both s and t are solutions of these equations. The 

 function ft satisfies a similar set of equations with —i in 

 place of i. 



The following expressions may now be adopted for the 



* H. Bateman, Amer. Journ. of Mathematics, April 1915. 

 t Johns Hopkins Circular, 1915 : Dissertation, Johns Hopkins 

 (1916). 



