﻿Cubic Crystals with Theoretical Atoms. 753 



and putting ja = rf^2> where c is the velocity of light, we find 



F 2 = + *^fi *' (GO' cos * + SS')R. • • (18) 



Denoting the ratio of the velocities of the charges to that 

 of light by yS and /3' respectively, resolving the force along 

 the three rectangular axes as before, and substituting the 

 value of R 3 as a function of the time, we find 



F 2 = + ^/3/3'A 3 (CC f cosa + SS'X%-mS + nS'cos«)(l + W )-%(19) 

 F 2 = + T ^/3/3'A 3 (CO'cos a + SS')(y*-mO + «C')(l + «)-V, • (20) 



F 2 = + ^/3/3'A 3 (CC'cos« + SS')(% + ^S , sin a )(l + w )-^. . (21) 



The two sets (13)-(15) and (19) -(21) are the complete 

 general equations for the instantaneous values of the electro- 

 static and magnetic components of the mechanical force 

 exerted upon the charge e by the charge <?', omitting the 

 third and fourth components as above mentioned, which 

 when averaged over a long period of time give zero. The 

 only quantities in these equations dependent upon the time 

 are the simple periodic functions S, S', C, and C', u being a 

 polynomial of seven terms each containing some of these 

 quantities. The expansion of (1-f w)~f into infinite series to 

 five terms is 



l-1.5u + l. 875u 2 -2. 1875^ + 2. 4609375m 4 -.... (22) 



Average Values. 



The process of finding the average values of these forces 

 over a long time T, integrating each equation with respect 

 to t between the limits of time and T, and dividing by T, 

 is to multiply each term of the series (22) by the quantities 

 in parentheses and integrate each separately, adding the 

 resulting integrals. It may be shown that the series (22) is 

 rapidly convergent for large values of #*, y #J and z%, due 



2 



to the factor — 2 in the expression for u (6) ; and experience 

 s 



has shown that there is no gain in including any terms 



above the sixth power of the distance. For this reason the 



next terms of (22) involving the fifth and higher powers of 



u are not required. It is evident, however, that the number 



of terms to be integrated arising from the fourth power of u 



Phil. Mag. S. G. Vol. 29. No. 174. June 1915. 3 C 



