and Transverse j\fass of an Electron. 735 



ellipsoidal electron of Lorentz the value of our integral is 

 (see Lorentz's 'Electrons,' p. 211) 



2 e 2 / r^yK 



e being the charge in electrostatic units, a the semi-major 

 axis of the ellipsoid, and c the velocity of light. Returning to 

 the axes of X, Y, Z, and putting co 2 = p 2 4- q 2 + r 2 , we at once 

 obtain for the components U, V, W of the momentum resolved 

 along the axes 



w 



2 P 2 ( r?4-n* + **^ -\ 



r 



3 ac 



i-z±r+r.\ 



Differentiating these expressions with regard to p, q, and r 

 respectively, and afterwards putting p = v, q = 0, r=0, we 

 obtain for the three masses the expressions usually given f, 

 the last two being; of course identical. 



r=0 



/?)W\ _ 2<? 2 / ^ 2 \-^ 

 \ dr JqZo 3ac 2 \ c 2 / 



r-0 



If, instead of the ellipsoidal electron, we take the conducting 

 spherical electron we of course obtain the well known 

 expressions corresponding to that case. 



* Lorentz uses the " Rational unit " of charge, which results in an 

 expression slightly different from the above. 



t In obtaining this expression the field of the electron at each point 

 in space is taken as the field corresponding to the steady motion of the 

 electron. All methods of determining the electromagnetic masses involve 

 this assumption. It is a very legitimate assumption for the purpose in 

 hand, because, as is easily shown, practically the whole momentum of 

 the field of the electron is contained within a space of the same order 

 of size as the electron itself, and consequently the field in this region 

 follows the motion of the electron practically instantaneously. It may 

 be noted that it is easy to show that even if the electron were not small, 

 the assumption would be justified for the case of 'the motion of an 

 electron starting from rest. " ' ' 



