228 Prof. A. Schuster on Electric Inertia and 



spherical surface of radius a, the magnetic energy is ^V/3a*, 

 so that the particle behaves as if it had a mass 2q q /3a. With- 

 out making any assumption, as to whether the magnetic forces 

 are to be taken as vanishing within the electron or not, we 

 may use the above expression, taking a to be a linear quantity, 

 not necessarily the radius of the electron but of the same 

 order of magnitude. If there are n particles per unit length 

 at a distance d apart so that nd= 1, the energy per unit 

 length will be <fu 2 /da, as far as the magnetic field established 

 by each particle is concerned. The mutual energy of different 

 particles has to be added in order to obtain the total mag- 

 netic energy. A pair of particles at a distance r from each 

 other will have a mutual energy of q 2 u 2 /r, and each particle 

 with its nearest neighbour on either side will therefore con- 

 tribute a term 2e q u 2 /d. Taking the remaining particles in 

 pairs we get for the mutual energy of a central particle and 

 p pairs on either side 



. The series S may be added up and the result expressed in 

 the form 



S = B + \ogp, 



where B is a number approximately equal to 0'577. 



W p is very large, the total magnetic energy per unit 

 length of the central portions of the row will be 



"^(2S + ^) = ^(2B + 21ogp + ^) . (1) 



where C stands for the current. 



3. I now compare this expression with that calculated on 

 the usual supposition, which is, that the electrification is not 

 confined to electrons but fills continuously a rectangular 

 space having a square cross-section with sides equal to d, 

 and having a length equal to (2p-f-l) d, for which we write 

 2D. The total magnetic energy in this case will be the same 

 as that of two linear conductors of the same length, and at a 

 distance apart which is equal to the geometric mean of the 

 square section r where 



r ='U7d. 

 For unit current in each of the conductors, the linear ele- 

 ments of which are ds x , ds 2 , the magnetic energy between the 

 conductor s 2 and the element dsi is 



* Heaviside, Phil. Mag. April 1889, p. 324. 



