242 Proceedings of the Eoyal Society of Edinburgh, [Sess. 
following letter of date July 8, 1916, written to Professor Whittaker from 
Blandford Camp ; — 
Dear Sir, — In reading rather hurriedly E. Cunningham’s The 
Principle of Relativity, I was much struck with the similarity between 
the relativity view of the electro-magnetic field and that of the Faraday- 
tube theory. I found it, in fact, very easy to write down expressions for 
the electro-magnetic field in terms of a system of lines of force in a four- 
dimensional space of the relativity type. As I have not noticed any 
reference to this matter anywhere (for though the method I have used is 
very much borrowed from Bateman and Cunningham, I think they seem 
to regard lines or tubes of force from a different point of view), I have 
taken the liberty of sending you my results. You will understand that I 
am not in a position to consult books just now, and so have no idea what 
has been done in this line previously. 
“ I supposed a system of moving lines of force to be determined by the 
equations 
= J ’ 
where a, /3 are real functions of the coordinates x, y, z, u { = ict) ; and m, n 
are parameters, to every pair of integral values of which corresponds a 
moving line of force. 
“ Now, on the theory of lines of foice, the component of electric displace- 
ment in any direction is the number of tubes which pass through unit area 
normal to that direction ; and the component of magnetic force is the 
number of tubes per unit time which cut unitdength in that direction 
(H = VqD, where q is the velocity). From these I find by taking the 
density of tubes in the elements 
dijdz, dzdx, dxdy, dxdu, di,du, dzdu, 
= etc., etc. \ 
" S(y, z) 
d{x, 'Ll) J 
( 2 ) 
“ Or, the electro-magnetic six-vector is the product of the two 
four-vectors 
From these follow at once 
di! dz c dt ' 
dx dy dz 
(3) 
