540 Mr. E. Cunningham on the 



!-' 



According to this transformation, as a mere geometrical 

 correspondence, the length of a line in the direction of the 

 axis of x moving with the axes A', as measured in the 

 coordinates x'y'z't', is greater than its length measured in 

 the coordinates x, y, z, t in the ratio 1 : (1 — v 2 /c 2 )*, so that 

 Lorentz's hypothesis of the reduction in the dimensions of a 

 body when it moves relatively to an observer is reduced by 

 this geometrical correspondence to the assumption that in the 

 variables associated with axes moving with it its shape is 

 unaltered — an assumption suggested by the fact that the 

 electromagnetic equations referred to those variables are 

 independent of the motion through the sether, and by the 

 attempt to form a purely electromagnetic theory of matter. 

 Thus if the single electron at rest has a spherical configura- 

 tion, and there are no other than electromagnetic forces, we 

 should expect it in motion to have a spherical configuration 

 when measured by the variables x'y'z 1 , which means that as 

 measured by the variables x, ?/, z it will have the spheroidal 

 shape as suggested by Lorentz. 



The electron as conceived by Abraham, on the other hand, 

 is spherical always as regards the variables x, y, z, and a 

 prolate spheroid as regards x'y'z' ' } the ratio of the axes being 

 1 : (l_^/ c 2)§. 



In either case the electromagnetic mass is defined as the 

 ratio of the external mechanical force on the electron to the 

 acceleration of the centre of the electron, and Abraham de- 

 velops two expressions for the longitudinal mass, i. e. for the 

 ratio of the force in the direction of motion to the acceleration 



in the same direction, viz.: — — and - ~— - for the case of 



av v dv 



the so-called quasistationary motion, G being the electro- 

 magnetic momentum and W the electromagnetic energy. 

 For the latter case these two expressions are proved identical, 

 but for the Lorentz electron they are not equal, and Abraham 

 deduces that W cannot be the whole energy of the electron. 

 But the fact is that in this case the mass as above defined is 



not equal to -—— ,this expression being obtained on the 



assumption that the electron is " rigid " (TJieorie der Elek. 

 ii. p. 155). If the change in the shape of the electron with 

 the change in velocity is taken into account, it will be found 

 that the mass as obtained from the change in momentum is 

 identical with the mass as obtained from the change in the 

 energy, as it clearly must be, since a quantity defined in a 

 perfectly definite manner cannot from consistent equations 

 be shown to have two different values. 



