1921-22.] The Faraday-Tube Theory of Electro-Magnetism. 243 
“ By choosing for a and /3 an appropriate form and by arbitrarily 
defining the positive direction of the tubes as that of the expressions (2), 
we could represent moving point charges, and from a discussion of their 
distribution in the four-dimensional space obtain the modifications in tlie 
kinematic scheme (3) due to the presence of electrification. 
The remaining (dynamical) scheme of four equations is of course 
deducible from the volume density or kinetic potential. 
‘‘ I noticed particularly that the velocity of the tubes was only defined 
(by the equations (1)) for directions normal to that of the tubes themselves, 
that is, only its component in the direction of the momentum is determinate, 
as in Cunningham’s theory of the stresses ; but, while on the simple sether 
theory only one velocity is considered at any point, when we are dealing 
with lines or tubes of force each separate set has its peculiar velocity, 
though the momentum corresponding to a tube is always perpendicular 
to it. 
With regard to electro- magnetic inertia, I have calculated the mutual 
electro-magnetic momentum of a pair of point charges moving obliquely to 
the line joining them, and find it not parallel to the velocity — as indeed is 
implied in Heaviside’s paper for low velocities {Phil. Mag., 1889). I do not 
know, however, if this point is of interest. Hoping you will pardon my 
troubling you so much, — I am, yours truly, W. G. Brown.” 
On August 1, 1916, Professor Whittaker, writing from Edinburgh, 
replied as follows : — 
‘‘ Hear Mr Brown, — Thank you for sending me an account of your 
results regarding lines of fprce. 
“ Probably you have noticed that your two functions a and /3 are connected 
with the old vector-potential and scalar-potential of the electro-magnetic 
field ; thus, if a.^,, ay , a^ are the three components of the vector-potential, 
and (p is the scalar-potential, so that the components of electric force are 
- 
1 dag 
c dt 
d<p 
dx ' 
etc., 
and the components of the magnetic force are 
da. 
dUy 
dz 
etc.. 
let us write down the ‘ differential form ’ or ‘ Pfaff s expression 
4- apz — cpdt . 
