﻿Absorption and Scattering of the a Hays. 905 



The logarithmic term in p x is the more important. f\ is 

 the sum o£ elliptic functions of all three kinds. 



In the case of a surface distribution of electrons the 

 corresponding integral reduces to 



P2 = 2^„ 2 {log(l + « 2 )-/ 2 (> s )J . . (5) 



where w 2 == a^v^/X 2 and 



f 1 x l l 2 dx 2 



This is an elliptic function similar to f\. For large values of 

 w, f\ and / 2 have limits 2/3 and 2 respectively and may be 

 expanded in inverse powers of w 1 ^, that is in the inverse 

 Velocity. For smaller velocities it is necessary to evaluate 

 them by means of Jacobi's functions. 



Before finding the equation of motion of the a particle we 

 must estimate the direct effect on it of the central charge. 

 The accurate equation (1) is applicable for this, with modi- 

 fied values of e and m. If these modifications are denoted by 

 accents, the loss of velocity due to the central charge is, 

 when small, given by 



A = 2k ' v X " 



v (HF) 2 V 2 +PV 



The cases of large deflexion and large change of velocity are 

 so rare that they may be disregarded, as they do not affect 

 the mean. The average loss of velocity from the central 

 charge thus is 



Now the electrical charge e' is equal to ne and hence 



so that 



2kv nk , ( * I /nk\ 2 



¥ 



n 



•iog-{i+*>/(J)\i+*7} 



Now nk/k' is the ratio of the mass of all the electrons to that 

 of the central charge, and this is quite a small quantity 

 Unless n is very large indeed ; and we shall subsequently 

 prove that this is not so. It is thus justifiable to neglect p 

 and take p as the mean loss of velocity in the atom. 



