114 Dr. L. Silberstein on the Electron 



aspect is ultimately reduced to finding appropriate integrals 

 oE Laplace's equation and adapting them to the surfaces of 

 the conductors, or, equivalentlj, to solving a linear integral 

 equation in which the unknown function appears under an 

 integral to be extended over the surfaces of the conductors. 

 In its electronic form the problem relates essentially to the 

 interior of the conducting bodies. No matter how rapidly 

 the density of charge decreases with increasing depth below 

 the surface, the problem is here, mathematically as well as 

 physically, a volume problem. 



2. Bv a fundamental theorem of the general kinetic 

 theory *, and by the well-known assumptions of the current 

 electron theory of metallic conductors, the number of free 

 electrons whose velocities and positions fall within the 

 element dvr = dudvdw of the velocity-space and within the 

 element dr = da dy dz of ordinary space occupied by metal 

 will, in electrostatic statistical equilibrium, be proportional to 



e-kd-'^.d^Jr (2) 



where k is as in (1), m the mass, c the resultant velocity of 

 an electron, and yfr, here an unknown function of #, ?/, z, the 

 potential energy of the electron in the resultant field of 

 force. Integrating (2) over the velocity-space, the number 

 of free electrons per unit volume will be 



c;.'-s*=cr.«.s(* + *\; ... (3) 



where e is the absolute value of the charge of an electron, (f> e , 

 as above, the given potential of the external field, andc^ the 

 potential of the (unknown) distribution of resultant charge 

 within the conductors. The constant factor C will be 

 determined presently. 



Let n be the number of free electrons per unit volume of 

 each conductor, in absence of the external field and in the 

 (macroscopically) neutral or un electrified state of the con- 

 ductors. Then, by the assumption of the theory, n is also 

 the number per unit volume of positively electrified atoms 

 (which will be assumed to be rigorously fixed), each carrying 

 the charge +e. Thus the resultant density p of electric 

 charge at a point a?, y, z, within any conductor of the system 



* See, for instance, J. II. Jeans' ' Dynamical Theory of Gases ' (191G), 

 p. 89, and passim, 



