VV=r.P ( n ) 



Theory of Metallic Conductors. 417 



The circumstance that there are three independent variables 

 instead of one or two does not, mathematically, make much 

 difference, so that (I) could be investigated in concrete cases 

 by the usual methods of the theory of integral equations. 

 That task, however, will be left to the specialist in this new 

 and promising branch of mathematics. Here it is enough 

 to state that, the square and the higher powers of pjne being 

 neglected, all cases of electrostatic distribution treated on the 

 lines of the electron theory will obey the linear integral 

 equation (I). 



Introducing, as before, the resultant potential (/> =• <j> e + fa, 

 that equation amounts to 



&p + <j> = (7) 



On the other hand, we have, by the very definition of <£, 



so that V 2 <£= r2 0> or to eliminate the auxiliary potential 



altogether, again by (7), 



1 

 L 2i 



This is a common partial differential equation * for p of a 

 form familiar from many chapters of mathematical physics. 

 It is a consequence of the integral equation (I), but does not, 

 of course, replace it completely. For in (II) every trace of 

 the given external field and of everything that concerns the 

 shape, the size, and the configuration of the conductors has 

 disappeared. In short, (II) is more general than (I). 

 However, although the equation (II) does not fully replace 

 (I), it may help us in solving (I), if we are not in the 

 position of solving it systematically by the methods of 

 integral equations. In fact, it is enough to find a sufficiently 

 general integral of (II) and to determine its more particular 

 form or its coefficients by substituting it into (I) and by 

 using the given total charges \ pdr i = q i '\. 



In order to explain the latter method and to illustrate, at 

 the same time, the meaning of the above general equations, 

 let us work out a pair of examples of the most simple kind. 



* The equation obtained by J. J. Thomson, loc. cit., is a special (one- 

 dimensional) case of the general equation (II). 



t Another way would be to attempt to supplement the differential 

 equation (II) by some plausible general surface-conditions. Such con- 

 ditions, however, would seem to be artificial from the point of view of 

 the electron theory. It is therefore that any conjectures about such 

 supplementary conditions are here omitted. 



