734 
MR. A. McAULAY ON THE MATHEMATICAL 
definitions. Equations (12), (13), (16) remain. Of these the last implies several 
other equations involving H and I. It may be left for the present. On the present 
theory equations (12), (13) are not true. 
63. It remains then to investigate the physical hearing of the points of difference. 
Equation (13), of course, could not be expected to represent the results of the present 
theory, from the definition of a current adopted in § 4 above. Equation (13) asserts 
that the dielectric current is 8d ' jdt. The question is by what on the present theory 
this statement must be replaced. Since c a flux = d, 
c' = m -1 yc = m~ 1 xd (my -1 d')/cfr = d' + m -1 y ~q t ( m X~ l ) d- 
Now by former paper equations (9), (11), 
m x ](o = ~ 
Hence 
~ (my -1 ) (o = — YVjVjjSo) p'lp'z = VV 1 xh r ojp\ [ibicl., equation (25)] 
= VxV^VVi = fnx-'YV'.Vvp’,, 
by Tait’s ‘Quaternions,’ 3rd. ed., § 157, equation (2). Hence 
c = d' + YV'jYd 'p\ = M'/dt -f YV'VdV - p'SV'd' ] 
C' = D' + YV'jY B' P \ = dV'/dt + YVYD'p' J 
(23), 
which equations have already been given m anticipation, in equation (38), § 50. In 
the case of an incompressible substance (solid or fluid) SV'p' = 0, and, therefore, 
o'= d' + Sd V'.’p .(24), 
and for a rigid body whose angular velocity (vector) is rj this simplifies further to 
c' = d' - Y^d'.(25). 
• 
Thus, on the present theory neither d' nor 8d7 dt is the dielectric current. 
The effect of the difference between the theories will be very slight in most 
experimental work, though it will, of course, lead to different results in the solution 
of certain problems which involve currents in moving bodies. 
64. There is one experimental result, however, in connection with which equation (23) 
has considerable interest. In the ‘Phil. Mag./ V., vol. xxvii [1889], p. 445, Professor 
