I Hi 

 I 



196 Mr. G. H. Livens on the Principle of 



But in all other applications ©f the general theory of 

 electromagnetism it is found more convenient to assume that 

 the magnetic energy, which for other reasons is regarded as 

 of kinetic type, is distributed throughout the field with a 

 density at any place expressed by 



where B is the vector of magnetic induction. 



Now, these two expressions for the magnetic energy do 

 not in the most general case agree, even in total amount,, 

 for, as Macdonald points out *, in the derivation of the one 

 from the other by the method of integration by parts, an 

 integral over the infinite boundary is brought in which is 

 not generally negligible. It thus becomes a question whether 

 the results of the dynamical theory can be used in the more 

 usual formulations and developments of the subject. An 

 attempt to justify such usage so far as concerns the 

 expression for the force on a moving charge has been made 

 by Larmor f by the examination of a special problem with 

 restricted conditions, but some doubt may still exist as to 

 the general validity of the argument thus employed. 



The question is settled in the theory of relativity J, where it 

 is verified that the usual expression for the force is consistent 

 with the second expression for the magnetic energy ; but 

 insofar as this verification is based on the differential invariant 

 theory associated with Minkowski's f our-dimensionai analysis 

 of the general theory, it can hardly be said to throw much 

 light on the physical bearing of the problem. 



It seems therefore desirable to attempt a direct formulation 

 of the general dynamical theory on the basis of the second 

 and more usual expression for the kinetic energy of electric 

 origin in order to confirm, if possible, the result obtained by 

 Larmor under special circumstances, which is generally used 

 without hesitation or restriction in either form of the theory. 

 The object of the present note is to show that such a formu- 

 lation can easily be effected and the results derived from it, 

 insofar as they are identical with those deduced on the older 

 basis, fully substantiate Larmor's conclusions from his special 

 problem . 



The principle of least action is applied, in the manner 

 already elaborated in full detail by Larmor, to determine the 



* L. c p. 33. 

 t L.M.S. Proc. 1915. 



X Cf. Cunningham, 'The Principle of Relativity' (Cambridge, 1914). 

 p. 158. 





