176 The Spheroidal Electron. 



If /3b = a, both expressions for M and both expressions- 

 for E lead to the same results : 



This is the case of the Lorentz electron. 



If we make a = b only the first pair of expressions can be 

 used as $b>a, and we obtain 



M = 



lbVac 



fv 2 + c 2 , c4-v__2cl 

 L v 2 * e — v v J ' 



C7ra \v c — v J 



which are the momentum and energy associated with the 

 Abraham electron. Both electrons are, therefore, particular 

 cases of the general spheroidal electron. 



The transverse mass is M/u, and well-known experiments 

 have been made to determine e/m and v, or, which is equi- 

 valent, e/M. and v. Thus it would, no doubt, be possible,, 

 though perhaps the mathematical work would be tedious, 

 to determine the value of the ratio of b to a for which the 

 theoretical value of e/M. would agree most closely with 

 the experimental results. A determination of this ratio 

 would bo of interest. We may, however, remark that 

 if the ratio of b to a tends to zero, the corresponding value 

 of M tends to 



e 2 /3v 

 16ac 2 ; 



that is, the aether momentum associated with an electron 

 whose shape is a plane circular disk moving with uniform 

 velocity in a direction perpendicular to its plane is equal 



to -^- times the momentum associated with the Lorentz 

 o 



electron moving with an equal velocity. The ratio of 



its transverse mass to its mass when v = is the same as 



for the Lorentz electron, and the experimental results 



could not decide between them. In the case of a very 



elongated prolate spheroid moving in the direction of its 



axis of symmetry, both the momentum and energy become 



very great. 



