FUNDAMENTAL FORMULATIONS OF ELECTRODYNAMICS. 225 



This last remark points to the possibility of obtaining an elementary deduction of 

 the expression 



for the complete electromotive forcive simply by calculating the rate of change of 

 intrinsic energy of a moving bi-pole, and the calculation has in fact been carried out 

 by LARMOR,* taking however a parallel plate condenser with equal and oppositely 

 charged plates, moving in a uniform magnetic field. 



An analogous argument in the magnetic case will give a deduction of the magneto- 

 motive forcive 



B- - k, E]. 







9. We have stated that the magnetic energy expressions just obtained are 

 effectively equivalent to those usually dei-ived, whereas as a matter of fact this is true 

 only of the final result ; the various formulae employed in the derivation of this result 

 are not in their usual form but it has been 'shown elsewhere t that they are consistent 

 with the complete dynamical theory, the more usual formulae and the various 

 modifications of them which have from time to time been suggested being all 

 inadmissible on this score. A complete discussion of the justification for this last 

 statement is necessarily beyond the scope of the present paper, but it may perhaps 

 serve a useful purpose if a brief outline of some of the more important reasons is 

 given, especially as they have some bearing on points raised elsewhere in the present 

 discussion. 



In the first place there is probably little or no difficulty in seeing the fallacy in the 

 usual and simplest form of the theory wherein the expression /xH 3 /87r for the magnetic 

 energy density is derived in the statical theory as potential energy and in the 

 dynamical theory as kinetic energy : we need only enquire as to the type of energy 

 represented by the same expression when the field is due in part to rigid magnets 

 and in part to steady currents. The more consistent result is obtained by taking 



as the expression for the density in the statical case as this agrees with the opposite 

 sign in the dynamical case and yet gives the same total. 



There is however another form of the results first tentatively suggested by HEKTZ 

 and HEAVISIDE and subsequently developed in great detail by other writers, more 

 particularly by R. GANS| and H. WEBER, wherein the difficulty presents itself in 



* ' Proc. Lond. Math. Soc.' (1915). 



t Cj. my ' Theory of Electricity,' p. 417, or ' Roy. Soc. Proc.,' vol. 93, A, p. 20 (1916). 

 J ' Ann. der Physik,' vol. 13 (1904), p. 634, and ' Encyklopadie der Math. Wissensch.,' vol. v., art. 15, 

 where references to other authors will be found. 



2 I 2 



