﻿Faraday-Tube Theory of Electro-Magnetism. 595 



as to the modification o£ 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 

 electromagnetic consequences of altering the Principle of 

 Action to be estimated, and partly to suggest plausible 

 directions for modification of the electromagnetic relations 

 themselves. It will incidentally be shown that the stress 

 which may be supposed to act in the electromagnetic 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 superfluous at present to specify any further 

 properties of the electric particles. 



The tubes at any point may be divided into sets dis- 

 tinguished by each set having a common direction and a 

 common velocity of translation. 



In what follows the vectorial notation of Heaviside is 

 employed *, and electrical quantities are measured in rational 

 units. Let the density of the tubes of the mth set and their 

 direction, at any point, be represented by the magnitude and 

 direction of the vector d nl ; then the number of tubes of that 

 set passing through unit area normal to the unit vector N 

 will be Nd w . 



Let D = 2d ;?i , (1) 



the summation including all the sets present at the point ; 



* [Heaviside's vector notation is a modification of Hamilton's quater- 

 nion notation, the main difference being that the quaternion product 

 of two vectors 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 9, where A, B are the lengths of A, B, and & 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 A 2 =— A 2 . 

 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 A. — C. G. K.] 



2Q2 



