﻿deduced from the Electrical Theory of Matter. 80 



t, from equations (19). The intersection of these surfaces 

 gives, of course, the contracted orbit which we expected, 

 and our investigation consequently shows us, that if the 

 electrons in the moving system move in the contracted 

 orbits we have assumed, and in the manner determined by 

 the investigation on pages 70-76, their mutual forces on one 

 another, brought about by these motions, are such as to keep 

 them moving in those orbits, without any external forces 

 being applied. 



The above investigation has involved the neglect of 



quantities of the order — ( p 2 V where W is the velocity of an 



electron relative to the system as a whole ; thus the results 



v . W 



are only true to the second order of ^ provided that — may 



be considered as a first order quantit}^ a result not likely to 

 be true in practice. The investigation is, however, true to 



any order of j y , provided that W is sufficiently small. It 



may be argued that, in limiting the investigation to small 

 values of W we are practically restricting ourselves to a 

 system in which there are no orbital motions, and that 

 consequently we achieve no more by our work than has been 

 achieved by the investigations on pages 66 and 67, for a 

 case in which the electrons in the system at rest are 

 absolutely devoid of motion ; the point is, however, that 

 directly we allow any finite value to the orbital motions, 

 however small, we are able to conceive our system as being- 

 one in secular equilibrium, and though no system with such 

 small orbital velocities may actually be found in practice, 

 we may look upon this system as one ideally possible. At 

 first sight a difficultv suo-gests itself, owino- to the fact that 



) :©S > w 



the smaller the value of v. the larger the value of — and 



' ° v 



apparently the more inexact our analysis, but we must realize 

 that as v is made smaller and smaller, the whole effect of the 

 translatory motion becomes more and more inappreciable, 



Wv 



and our analysis is always true to the order -yr 2 • 



As has been remarked on page SI, if quantities of the 

 order of f — ) ( p-A are no ^ negligible, the orbital velocities of 



the individual electrons will make their appearance explicitly 

 in our expressions for the effects of imparting the translatory 



