ON ELECTRICAL THEORIES. 135 
As it is in the difference between these equations that the difference 
in the theory really lies, it will be instructive to look at them from 
another point of view. We know of no way in which the quantity of free 
electricity can be altered except by electricity being conveyed by con- 
duction currents to the place where the alteration takes place. Assuming, 
then, that the alteration in the density is caused by such currents 
ap 07. Oe 
da* dy? da dé? 
de 
l (pi —p2) +m (41-2) #2 (1-72) =F 
So that Helmholtz’s equations taken in conjunction with these are 
equivalent to the condition 
dz dy 4d 
det dy ur 
L (¥;—Z2) +m (1 —- Do) +” (31-32) =0. 
Thus on Helmholtz’s theory the dielectric currents behave like the 
flow of an incompressible fluid, while on Maxwell’s theory it is the total 
current, which is the sum of the conduction currents and the dielectric 
currents which behave in this way. 
The equations we have arrived at for the dielectric currents seem 
inconsistent with Helmholtz’s definition of them ; for since 
=0; 
x=eX, 
with similar equations for p and 3, and since in a medium at rest 
where U, V, W are the components of the vector potential. If we consider 
a surface separating two portions of the same dielectric and coated 
with electricity whose surface-density is o, we have, since U, V, W are 
not discontinuous on crossing the surface, 
5 Punks i ; £4 fn df, d d do72 
l a? sts i tO (31-3 = —*§ [' at” ay* Si al 
where [2 uf +m za +n Z| denotes the difference between the values 
dx dy dz Jy 
d d 
of 1 2 +m at <on the two sides of the surface. 
do do do]? 1 
= [? dz ™ dy t™ i ee % 
do 
so that 1 (%)—¥2)-+m (Bi—Ba) +" Gi) =] ap 
and so cannot vanish if the surface-density of the electricity changes ; 
