﻿Faraday-Tube Theory of Electro-Magnetism. 597 



The meaning attached to the above quantities is that if 

 we write 



m 



(T-U>*r, 



where the volume integral is extended throughout all space, 

 then L may be used as the Lagrangian function in equations 

 of motion of the usual form. For the sake of brevity, 

 vectorial general coordinates will be employed. In order to 

 preserve the form of the equation 



dt'"dg "dq ' 



it is sufficient to write, in the case of a vector coordinate r 

 (equivalent to the three scalar coordinates x, y, z), 



9 .■*+•* +J a (■ (9) 



or dx J ^y d~ J 



This notation in vectorial analysis is of course not generally 

 applicable, but is convenient for the purposes of the present 

 paper. The general results of differentiation which will be 

 required are 



^ s as = a . (10) 



gs^s = 2ts, (11) 



where s is any vector variable, a is a constant vector, and i/r 

 is a constant self-conjugate linear and vector operator. 



4. To define the general coordinates, let all tubes at a 

 given moment be divided into small unit lengths ; and let r 

 be the vector from a fixed origin to the centre of one such 

 unit segment, which forms part of a tube of the mth set, then 

 the Lagrangian equation corresponding to r will be 



i|5-|5!-0 (12) 



dt dr or 



Now, when a unit length of a tube of the mth set is added 

 to, or removed from, an element of volume, the increase or 

 decrease of the whole Lagrangian function due to this 



