FUNDA>rEXTAL FORMUT.ATTOXS OF ELECTRODYXAMTCS. 
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 he taken as kinetic or- potential energy. One might 
pei’haps infer that as tlie 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 
I r^’ 
- ^ J cZN 
c J 
suggests that this part of the energy at least is kinetic energy. The difficidty 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 
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 
Jdyj 
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 enei'gy 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 troidde. To effect the transformation from the expression 
,-pm 
dv j 0 dp,„ 
dy j'"(H dio) 
the method of integration by parts is employed. But Lakmor 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 
to the equivalent expression 
