218 ME. G. H. LIVENS ON THE 



In the main the details of the variational calculation possess no novel features and 

 need not here be elaborated. There are, however, one or two terms which require 

 careful handling, especially when finding the variations due to the alteration of position 

 of the matter. 



The variations of terms of the type 



due to variations in the position co-ordinates of the elements of matter to which the 

 vectors P and r m are attached should not be performed until the differential operators 

 affecting the P function are eliminated by an integration by parts. If we bear this 

 in mind we shall find that the final result for the variation consists of terms at the 

 time and space limits, which require separate adjustment, together with 



f'dt \dv \L + -jL /SB, B- i 

 J(, J L 47r \ c 



- (SB., Curl A : - - d -^} - (P, (r m V) ( V^+ i ^ 

 47r\ c dt J \ \ c dt 



- l -Ur m , [Curl A 1( ^1) - l - (r m [curl ^, 

 c \ i_ at _j/ c \ (M 



n i A 



CurlA,, _ 



' e , V0 -| [r m + r e , Curl AjJ 



. + (_V^-I^Ai + i[r m , CurlAj, ^P 



\ S* f^lf f* "- 



1 r, 



where in the terms (Sr m V) (Curl A,, [Pr m ]) and (<5r m V) (Curl A 2 , [Ir m ]) the vector 

 operator V (whose components are ~ , , } is presumed to affect only the 



\ rtrr. t\ii r\v 



dx dy 

 functions A, and A 2 . 



The variations r m , r e which determine the virtual displacements of the electrical 

 and material elements and the variations E, B, P, ... , can now be considered as all 

 independent and perfectly arbitrary, and hence the coefficient of each must vanish 



