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 electi’omagnetic origin acting on the 
elements of a polarised medium still seems to be the subject of some doubt. 
Maxwell! 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.| 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 betw'een 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 
* Cf. Larmor, ‘Eoy. Soc. Proc.,’ vol. 71 (1903), “On the Mechanical and Thermal Eelations of the 
Energy of Magnetisation.” 
t ‘ Treatise,’ II., Ch. II. 
X Cf. Leathem, ‘Eoy. 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 conwncing. 
§ 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. 
