﻿518 Mr. S. B. McLaren on Emission and Absorption 



F and cf> are the vector and scalar potential, tl the vector 

 velocity of the charge whose density is p at any point. 

 Regarding (13) as a condition imposed upon F and <p (11) 

 and (12) can be deduced by making the variation of the 

 action zero. 



B$$Ldvdt=0 (14) 



i =i6f +v ^'-sl^- 1F ) 2+ ''( ! ?-*)^-( 15 ) 



In (14) dv is any element of volume in the field. L m in 

 case it exists is the Lagrangian function of material origin. 

 It may be shown that the principle of relativity allows L m to 

 be any function of p 2 (l — U 2 /c 2 ) but this is not used in what 

 follows (L is invariant under a Lorentz-Einstein substitution). 

 The equations of motion of the electric charge can be inferred 

 from (15) combined with the equation of continuity 



d £+J)iv.pU = (16) 



Write F = F 1 + F + V%, (17) 



where Div. F x = (18) 



Div. F = (19) 



X and F will be so chosen as to make (18) and (19) possible. 



(17) gives 



V'-'x = Div. F = - ±® by (13), (18), (19), . . (20; 



ves^)=(v-is^ : ^b y( i 2 ). 



Hence if -%+$ = 4>*> (21) 



c at 



then V 2 ft> + W = ° (22) 



So that O is the potential of the charge p. I suppose the 

 volume V containing the radiation to be bounded by a 

 surface S as far distant from the radiating matter as we 

 please. (f> is to be so chosen as to vanish at S. which will be 

 supposed perfectly conducting. 

 From (20) 



i^-^t^'cd^J^)--rat • W 



