﻿904 Mr. C. G. Darwin : A Theory of the 



We shall first find the result supposing the electrons distri- 

 buted uniformly inside a sphere o£ radius o^ and afterwards 

 supposing them arranged over the surface of a sphere of radius 

 o- 2 . The numbers 1 and 2 subscribed will refer throughout to 

 these alternative hypotheses. 



Let P denote the distance from the centre of the atom of 

 the initial line of motion of the a particle. In passing through 

 the atom this line is slightly changed and the velocity is 

 reduced, but the effects are small, and no error is produced 

 by supposing the velocity on approaching each electron equal 

 to the velocity of approach to the atom, and by supposing 

 the line of motion through the atom to be straight. Let the 

 position of an electron be denoted by cylindrical polar coor- 

 dinates r, (/>, z. The loss of velocity due to an electron at 

 r, 0, z is 



2kv \ K + F + r 2 - 2Fr cos <f>, 



and the chance of there being an electron at this point is 



/4 

 -7TG-! 3 . rdrdfydz. 



The whole loss of velocity on the average thus is : — - 



This expression is to be averaged for all values of P. The 

 result gives p x , the mean loss of velocity of a particle in 

 traversing an atom. Then :— 



•J \ 



Three of the integrations can be performed, and if 

 (TiV^j^ — Wi, the expression may be reduced to 



Pi^2^« 1 {log(l + ttl )-/ 1 ("-'i)} • ■ (4) 



where 



