2S4: Doppler's Factor and the Electron Theory. 



dimension of the electron parallel to the radius vector is 

 modified in the desired ratio is a matter of controversy. In 

 effect, this change of dimension is by no means j ustified by the 

 subdivision of the electron into its elements of volume, each 

 of which must correspond to an instant of emission, giving a 

 slightly different value to the radius vector. This integration 

 throughout the volume of the electron occurs in determining 

 the mean radius vector, which only differs from the radius 

 vector of the centre by a quantity of the second order. 



This may be shown as follows in the case of rectilinear 

 motion. On taking P as origin and the velocity it as parallel 

 to the ^-axis, we find, by eliminating t — t, the radius vector 

 at the instant of emission expressed as a function of the co- 

 ordinates of the electron at the instant t . This expression, in 

 which s is equal to (l—u 2 /v 2 ), is as follows *: — - 



-[>+ vv+-) S+ .4 



This corresponds to the centre of the electron ; it is 

 required to find the variation of r due to the variation of the 

 •co-ordinates, and for this purpose the above expression is 

 differentiated with respect to a?, y, z, and then dx, dy, dz are 

 replaced by oc' ', y', z f , which represent the co-ordinates of a 

 point in the interior of the electron relatively to its centre. 

 This gives 



di 



1 rr» , ;n , si/ , sz ,~] 



#/D, y/D, z/J) are very nearly equal to the cosines of the 

 angles made bv the radius vector with the axes, since D is equal 

 to \/ (y 2 + z 2 )s + a: 2 , which is equal to the radius vector at t if 

 we suppose s equal to 1. It is required to find the mean 

 variation of dr for all points of the volume of the electron ; 

 and this is done by integrating the above expression over the 

 •entire volume and dividing by the volume. Now each term 

 when integrated with respect to the variable which it contains 

 gives a result which is a square and has the same value at the 

 two limits + 1 and — I which represent the dimensions of the 

 volume ; the integral therefore vanishes, and the mean radius 

 differs from the central one only by an infinitesimal of the 

 second order. This is not so if r is multiplied by K, since 

 ujv itself is not even necessarily a very small quantity of the 

 first order. 



The hypothesis which I propose as complying with plausible 

 conditions consists in supposing that the action of the electron 

 in motion is due to an oscillatory state of the aether, which is 



* Bucherer, Evnfiihrung in die Elektronentheorie, p. 84. 



