Doppler's Factor into the Electron Theory. 281 



the volume integral being extended throughout infinite space. 

 Hence, by (I. a) and (II. a), 



t=tc,- 



and a similar expression may be obtained for A. In accord- 

 ance with (6), each volume element of the integral should be 

 considered at the instant t defined by the relation 



r 

 t= t —. 



V 



Let us consider the nature of the variable t defined by (6), 

 t being regarded as constant. Let the whole of space be 

 divided up by spherical surfaces of radius r, r varying from 

 zero to infinity, corresponding to values of t varying from t 

 to minus infinity. I call t the time of emission ; and it is 

 evident that to a positive increment dr there corresponds a 

 negative increment vdt, since all the emissions are considered 

 with respect to the same instant t at P . 



On the other hand, Maxwell's equations and their trans- 

 formations are" directly applicable to points such as P of the 

 field, which may be termed the transmission field, where 

 the function cf> and its derivatives, whether with respect to 

 time or with respect to the space variables, are the result of 

 the propagation of these quantities from the emitting source. 

 Since our equations apply to the point P , it is necessary 

 to consider an elementary variation of t and of the position 

 of P , the differentials concerned being denoted by d Q , 

 dscoi dy , dzQ. 



Regarding both t and t as variable in (6), we have 



dr = v(dt —dt) (7) 



Let E and E' be the two positions of the electron at the 

 instants t and t + dt of emission. The path EE' described by 



