1921-22.] The Faraday -Tube Theory of Electro-Magnetism. 225 
XVI. — On the Faraday-Tube Theory of Electro-Magnetism. By 
the late William Gordon Brown. Communicated by Dr C. G. 
Knott, F.R.S., General Secretary, along with a Biographical Note 
of the Author. 
(Read January 9, 1922.) 
1. The method of describing a field of force by means of lines or tubes of 
induction, which originated with Faraday, was given a quantitative form 
by Sir J. J. Thomson,* and further discussed by N. Campbell in his book 
Modern Electrical Theory. Since Maxwell himself looked on his work 
as a mathematical theory of Faraday’s lines of force, one is tempted to 
examine the original physical theory for hints as to the modification of 
the Maxwellian theory to suit certain modern requirements. 
What is attempted in the present paper is a reconstruction of the 
quantitative theory of Faraday tubes on a dynamical basis from the 
minimum of hypotheses : partly to enable the electro-magnetic consequences 
of altering the Principle of Action to be estimated, and partly to suggest 
plausible directions for modification of the electro-magnetic relations them- 
selves. It will incidentally be shown that the stress which may be 
supposed to act in the electro-magnetic field requires certain modifications 
if the theory of lines of force is adopted. 
2. The first assumption required is as follows; — A tube of induction, or 
Faraday tube, may be defined as a continuous line having certain physical 
properties. Any tube may either be a closed curve, or its ends be connected to 
a positive and a negative electric particle respectively ; the positive direction 
will then be from the positive to the negative particle. It would be super- 
fluous at present to specify any further properties of the electric particles. 
The tubes at any point may be divided into sets distinguished by each 
set having a common direction and a common velocity of translation. 
In what follows the vectorial notation of Heaviside is employed,f and 
* Recent Researches, chap, i ; Electricity and Matter, chap. i. 
t [Heaviside’s vector notation is a modification of Hamilton’s (piaternion notation, the 
main difference being that the quaternion product of two \ectors AB is not used in 
Hamilton’s sense but is used to mean the scalar of the complete product — that is, Heaviside’s 
AB is equivalent to Hamilton’s -SAB, and may be defined geometrically as equal to 
AB cos 0, where A, B are the lengths of A, B, and 0 the angle between them. As in other 
non-associative vector algebras, the square of a vector is equal to the square of its length ; 
in quaternions The notation introduced by Gordon Brown in equations (9), 
(10), etc., has been suggested by others but generally discarded. Burali-Forti and Marcolongo, 
however, make it a feature of their system of vector analysis. As a notation it is misleading 
as an operator it is inferior to the quaternion v. — C. G. K.] 
VOL. XLII. 
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