696 
ME. A. McAULAY ON THE MATHEMATICAL 
standard position of matter, obeys the laws of incompressibility that round every 
circuit there is an electromotive force equal to the rate of decrease of the surface 
integral of 47r H V/ through the circuit, and that K VZ — H/47T appears in subsequent 
equations in such a manner as to compel us to identify it with the magnetic moment 
per unit volume .* It is clear, then, that 47r H V/ is, according to the present theory, the 
magnetic induction. As the theory is developed below it is convenient to define B 
as equal to 477- H V/ and call B the magnetic induction, leaving the justification till we 
examine the detailed consequences of the theory. It is well to insist on this result 
here, as it does not appear obvious in the work below, but only comes out when a 
general review of a great part of the paper is made. To put the matter in the form 
of a proposition :— 
If the two fundamental assumptions are made —(1) that 47tC = A r VH, and. (2), that 
l, the Lagrangian function per unit volume, can be expressed in terms involving H 
but independent of magnetic induction and of magnetic moment per unit volume, then 
the magnetic induction must be = 47t h V?. 
17. The other most important features of the fundamental assumptions are first 
those already described with reference to the electric coordinates, and the expression 
for the current in terms of the displacement ; and secondly the manner in which are 
treated the two currents, conduction and dielectric (the latter being inappropriately, 
on the present theory, denominated the “ displacement current”). If there are (and 
physicists seem agreed on the point) two independent currents whose sum appears in 
the equation 47 tC = VVH, and whose sum obeys the laws of incompressibility, it seems 
to me of the nature of a truism that there must be also two independent electric dis¬ 
placements whose sum obeys the laws of incompressibility. I therefore, from the very 
beginning, recognise two displacements, d and k, which 1 call, for want of better 
names, the dielectric and conduction displacements.t This naturally leads to the 
contemplation of two independent kinds of electromotive force. This last, however, 
is subsequently satisfactorily disposed of. 
18. Before leaving the fundamental assumptions, let me remark that though in some 
important respects the present theory may seem to differ from Maxwell’s, it will be 
found, I think, that just where the difference seems to be most marked, is Maxwell’s 
theory most vague. All the differences, if they really be such, have been forced on 
me unwillingly in the attempt to put into definite form what I take to be the essence 
* Strictly speaking, the last clause should be modified by the condition “ if the present position be 
taken as the standard position.” This, however, is only an accident of the particular form of enunciation, 
which, at the present stage, is unavoidable. 
t Perhaps it would be better to call them the elastic and frictional displacements or the reversible and 
irreversible displacements. I wish to leave this point open for those better qualified to decide. Of the 
three sets of terms suggested above, the last seems to be the best. The only reason for adopting in the 
present paper the names given in the text is to imply the origin of the assumption that there are two 
such displacements. Of course, if we call the two displacements reversible and irreversible, we must 
also call the corresponding currents reversible and irreversible. 
