﻿Faraday- Tuhe Theory of Electro-Magnetism. 601 



rendered necessary by the purpose of the present paper, 

 which is not to deduce the properties of the tubes from the 

 known laws of electromagnetism, but to show that, given 

 the tubes with the (essential) properties assigned to them by 

 Sir J. J. Thomson, the laws of electromagnetism follow. 



5. It remains to discuss the forces acting on the electric 

 particles. Referring to the figure on p. 598, let B be a 

 particle at the end of the tube B, C, D. Then the change in 

 L due to the displacement of the end of the tube from B to 

 A (introducing a new segment BA), is by (13) 



SL = 8r(E + VqJB), (2S) 



since 



Sd wl = AB=-Sr, 



B being the positive end of the tube, and thus equivalent to 

 a positive unit of electricity. Hence the force acting per 

 unit charge moving with velocity q is 



F = E-f-VqB, (29) 



the Fifth Law of Electromagnetism. 



6. The definite dynamical assumptions of this theory 

 enable us to examine very thoroughly such questions as the 

 stress in the field and the mechanism of radiation. 



Heaviside * has given a general discussion of the problem 

 of stresses from which it is not difficult to deduce the 

 following general result : — 



Let yjr be the operator of Maxwell's stress, 



^ = E.D + H.B-i(ED + HB), . . . (30) 



where any vector operand forms with D and B scalar products 

 in the first and second terms. When this operand is a unit 

 vector N, -v^ N is the stress on the plane perpendicular to N. 

 Let -v/r be the stress derived from \jr by putting for E, 

 E4- VqB, and for H, H — VqD, namely 



^ = i|r +- VqB . D- VqD . B-i(VqB)D + ^(VqD)B 



= f + VqB . D + VDq . B— DVqB 



= to + VDB.q (31) 



by mere vector transformation. 



Then if N is unit normal to a surface moving with a velocity 

 q at any point, \/rN is the flux of momentum through the 

 surface in the direction opposite to the positive direction of 

 N, per unit surface per unit time. 



* ' Electrical Papers,' vol. ii. pp. 521 et seq. ; also Phil. Trans. A. 1892 



