1921-22.] The Faraday-Tube Theory of Electro-Magnetism. 231 
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 + VqB, (29) 
the Fifth Law of Electro-magnetism. 
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 i/jq be the operator of Maxwell’s stress, 
iAo = 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, is the stress 
on the plane perpendicular to N. 
Let ip be the stress derived from by putting for E, E-f-VqB, and 
for H, H-VqD, namely 
i/, = + VqB . D - YqD . B - VqB)D + 1 ( VqD)B 
= if/Q + VqB . D + VDq . B - DVqB 
= ,Ao + VDB.q (31) 
by mere vector transformation. 
Then if N is unit normal to a surface moving with a velocity q at any 
point, ipN is the flux of momentum through the surface in the direction 
opposite to the positive direction of N, per unit surface per unit time. 
To see that this is true we have only to apply the theorem of 
divergence ; in the first place we note that since 
^ = Vd„B, (18) 
summing for all value of m we have VDB equal to the momentum per unit 
volume. But 
■AoV = |vDB, (32) 
a result easily deduced (Heaviside, loc. cit.) from the circuital laws, and 
usually expressed in words by stating that Maxwell’s stress gives rise to 
a translational force per unit volume equal to the rate of change at a fixed 
point of the momentum per unit volume (the absence of electrification 
being assumed). We are thus entitled to say that ip^ is the flux of 
momentum per unit area of a fixed surface. Now it is clear that VDB . qN 
* Electrical Papers^ vol. ii, pp. 621 ef seq. ; also Phil. Trans.., A, 1892. 
