﻿Electron Theory of Matter. 23 



The notation is the usual notation of Bessel functions. 



The function U is of special importance for our purpose. 

 The series converges when ft < 1, very rapidly for large values 

 of n and small values of ft. In this case it is sufficient to use 

 the first term only. Whatever the value of ft, U is positive. 

 The equations of motion become 



^tt p 2 d(mv) 



U ~ e 2 dt 



(1) 



Pi-^(1+/3 2 )K+^V='^. . . . (2 ) 



§ 4. Equation (1) shows that a strictly steady motion is 

 impossible when we restrict ourselves to the case of an 

 invariable electron and exclude all forces not of electro- 

 magnetic origin. For given values of n, ft, p it determines 

 ft. Equation (2) determines p when ft is given, but ft itself 

 remains arbitrary. This is the difficulty first pointed out by 

 Jeans (§ 1). There is nothing to fix the constitution of the 

 atom, if it be supposed built up of invariable electrons in 

 orbital motion, with only electromagnetic forces between 

 them. 



The difficulty can be overcome by supposing the electron to 

 vary slowly in radius. 



In this case, 



dm a e 2 . , n . a , c ON 



Equation (1) becomes 



nTJ__ p*+(0) d 

 ft ~ C a 2 W 



This equation together with (2) determines ft and p when n 



and ~2 are given. Since U is essentially positive a must be 



positive. 



We easily find on examination that the right-hand member 

 is of order 10 -24 at most for the negative electron, if 



- =10 -10 ; probably it is very much less. The factor p 2 ^r(ft) 



depends on ft, but does not vary very rapidly with it ; the 

 function U on the other hand, for fairly large values of n, 

 varies very rapidly with ft ; in consequence an approximate 

 knowledge of p 2 ^(ft) is quite sufficient to allow us to calculate 



ft accurately when n and - 2 are given. When ft is known p 

 is calculated by (2). a 



