208 ME. G. H. LIVENS ON THE 



derived in the process of averaging the minute current whirls into their effective 

 representation as a distribution of magnetic polarity.* 



Further, the expression for the forcive of electromagnetic origin acting on the 

 elements of a polarised medium still seems to be the subject of some doubt. 

 MAXWELLt derives an expression in the magnetic case by statical considerations 

 based on the method of energy, and then seems to regard it as generally valid under 

 all circumstances. Objection has however been taken to MAXWELL'S expression 

 by certain writers who, basing themselves on the presumed analogy between the 

 dielectric and magnetic cases, prefer a form of expression differing from MAXWELL'S 

 by a quantity which vanishes in the statical case considered but which is of funda- 

 mental importance in the derived problem of reducing the general forcive of 

 electrodynamic origin to a representation by means of an imposed stress system. It 

 appears in fact that the presence of this extra part in MAXWELL'S expression is 

 effective in securing the ordinary expression for the subsidiary term arising in the 

 induction, which has given rise to the conception of electromagnetic momentum, on 

 account of its being a perfect time differential. In the alternative form of the theory 

 the perfect time differential is not secured so that the idea of electromagnetic 

 momentum is lost.J In his edition of MAXWELL'S treatise, J. J. THOMSON adds a note 

 attempting to justify MAXWELL'S form of the expression, but his discussion can easily 

 be shown to be erroneous, for he fails to distinguish between the true and complete 

 currents of the theory, the latter containing a constituent, viz. , the rate of change of 

 sethereal displacement, which is not affected by the magnetic part of the complete 

 electromagnetic forcive ; nevertheless, the later discussions of the question from the 

 point of view of the theory of electrons have confirmed MAXWELL'S original expression 

 for the magnetic forcive, but they apparently still give the alternative expression for 

 the dielectric case. 



It was with the view to clearing up these and certain other difficulties that the 

 present discussion was undertaken, the object aimed at being the formulation of a 

 complete and precise statement of the theory in the only form in which it is logically 

 consistent, then to compare this form with current statements of the theory, and 

 finally to exhibit in their true aspects the various derived theories which are included 

 in the general scheme. The original differential theory will be linked up with the 

 subsequent dynamical theories by a discussion in its most general form of the derivation 



* Of. LARMOK, 'Roy. Soc. Proc.,' vol. 71 (1903), "On the Mechanical and Thermal Relations of the 

 Energy of Magnetisation." 



t ' Treatise,' II., Ch. II. 



I Cf. LEATHEM, ' Roy. Soc. Proc.,' A, vol. 89 (1913), p. 34. In this note Mr. LEATHEM attempts to 

 avoid the discrepancy by adding a new term to the force in the elementary polar theory. His only 

 argument in favour of this force is that it overcomes the difficulty, so that it is not very convincing. 



A complete statement of the fundamental results of the theory so far as they existed up till 1916 is 

 given in my treatise, ' The Theory of Electricity ' (Cambridge, 1918). The present paper may be regarded 

 in some measure as a correction and generalisation of the statement there given. 



