436 The Theory of Aether and Electrons in the 



analytical theory of electrons, nothing more would be required 

 than to modify the formulae by writing e (the charge of an 

 electron) in place of pdxdydz. That this is not the case was 

 shown* a few years after the publication of the Versuch. 



Consider, for example, the formula for the scalar potential 

 at any point in the aether, 



where the bar indicates that the quantity underneath it is to 

 have its retarded value, f 



This integral, in which the integration is extended over all 

 elements of space, must be transformed before the integration 

 can be taken to extend over moving elements of charge. Let 

 de denote the sum of the electric charges which are accounted 

 for under the heading of the volume- element dx'dy'dz in 

 the above integral. This quantity de is not identical with 

 ~p'dx'dy'dz'. For, to take the simplest case, suppose that it is 

 required to compute the value of the potential-function for the 

 origin at the time t, and that the charge is receding from the 

 origin along the axis of x with velocity u. The charge which 

 is to be ascribed to any position x is the charge which occupies 

 that position at the instant t - x/c; so that when the reckoning 

 is made according to intervals of space, it is necessary to 

 reckon within a segment (x 2 - a?i) not the electricity which at 

 any one instant occupies that segment, but the electricity which 

 at the instant (t - xjc) occupies a segment (x* - x\\ where x\ 

 denotes the point from which the electricity streams to x l in the 

 interval between the instants (t - x z ,'c) and (t - x^/c). We have 

 evidently 



&'i - %'\ = u (%2 - %i)/c, or x z - x\ = (x 2 - #0 (1 + u/c). 



For this case we should therefore have 



I^ ?' dx'dy'dz' = (l + ^'dx'dy'dz'. 



Xi \ cj 



* E. Wiechert, Arch. Neerl. (2) v (1900), p. 549. Cf. also A. Lienard, 

 L' Eclairage elect, xvi (1898), pp. 5, 53, 106. 

 t Cf. p. 298. 



