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XVI. The Energy in the Electromagnetic Field. 



To the Editors of the Philosophical Magazine. 



The University, 



T\-a> a -d Qtdo Sheffield. 



VEAK SIRS,— Noy 6th? 191g 



Ij" N your issue of August, 1917*, Mr, A. E. Biedermann 

 JL raises certain objections against my criticism of his note 

 on the " Energy of the Electromagnetic Field," on the score 

 that this criticism does not in fact dispose of the fundamental 

 difficulty which necessitated his modification of the usual 

 expression for the magnetic energy density. 



The main point at if?sue seems to be whether the extra 

 term in the magnetic energy, viz- 



A being the vector potential, is zero or not in the cases with 

 which Mr. Biedermann deals. I asserted in fact that it was 

 zero, without offering a proof of the statement, but he still 

 contends that this assertion cannot be justified. However, 

 he agrees that if it is possible to prove that a certain double 

 line integral vanishes, then his contention falls to the ground 

 and there is then nothing in his modification. This integral is 



8 



COS 6 



asi as 2 



taken round any two closed curves in space, a 1? a 2 , € being 

 the angles the elements ds^ ds 2 make respectively with the 

 radius r joining them and with one another. 

 Now it is easy to provef that 



cos e — cos a 1 cos a 2 = r- 



d 2 r 



>! ds$ 



and thus the integral is 



\ ] 7 . jrds x dsc. 



JJds 1 ds 2 



and obviously vanishes. 



I am, 



Yours very truly, 



G. H. Livens. 



* Circumstances over which I have no control have prevented my 

 answering- this letter before. 



t The details of the calculation are given in Poincare, Electricite et 

 Optique (2nd Ed. Paris 1901), p. 233. 



