FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 227 



function, whilst the force on a current is to be obtained as the positive gradient with 

 respect to its position co-ordinate. 



Unfortunately all the authors concerned merely talk of magnetic energy without 

 specifying whether it is to be taken as kinetic or potential energy. One might 

 perhaps infer that as the results are interpreted in terms of a static potential function 

 it is implied that all the energies are potential, but the fact that the forces on the 

 currents are derivable as the positive gradients of the function 



Jc?N 



suggests that this part of the energy at least is kinetic energy. The difficulty of 

 sign is therefore still present. 



Even if we confine ourselves to the statical theory the same interpretation is not 

 entirely free from difficulties of another kind. The potential energy in the field is 



taken to be represented by 



r o> 



<j> dp,,,, 



but this expression really represents the total energy in the field ; in the general case 

 the only part of this energy which is mechanically available is 



r r* 



j do j Pm d<j>, 



and this is properly speaking the potential function from which the mechanical forces 

 acting on the magnetism are to be derived. Of course, when the law of induction is 

 linear the intrinsic energy of the field is equal to the available energy, but even then 

 their natures are fundamentally different and equality in their magnitudes is hardly a 

 sufficient justification for confusing the one with the other. 



Apart from this difficulty, however, the next step employed in the development of 

 the theory will cause some trouble. To effect the transformation from the expression 



to the equivalent expression 



the method of integration by parts is employed. But LARMOR has shown that two 

 expressions of this type being derived the one from the other by the method of 

 integration by parts, really represent fundamentally different distributions of the 

 energy in the field, although the total amounts represented by them are the same. 

 The two expressions cannot therefore be used indiscriminately to determine the 

 stresses around an element of the magnetic matter. It is not, of course, possible at 



