ON ELECTRICAL THEORIES. 139 
theories; and, in fact, make as many assumptions about the constants as 
we may, there are still differences between the theories. 
In order to get as general a theory of these dielectric currents as 
possible, we shall investigate the consequences of assuming merely that 
these currents are proportional to the rate of change of the electromotive 
force, and write dielectric current=y (rate of change of the electromotive 
force), where 7 is a constant which for the present is left indeterminate; 
Tn Maxwell’s theory »=K/47, where K is the specific inductive capacity 
of the dielectric ; in Helmholtz’s theory, 7 is also proportioual to the spe- 
cific inductive capacity. We shall denote the components of the dielec- 
tric currents by the symbols f, 7, h; the components of the conduction 
current by p, g, 7, and the components of the total current by wu, v, w, so 
that 
u=p roy? 
du . dv , dw 
— f= ee | 
de dy dz i 
L (u,—ug) +m (vj) — v2) +0 (W,—w2) =; 
on Maxwell’s theory { and & are each zero. 
If F, G, H are the components of the vector potential, then by 
y. Helmholtz’s investigation of the most general expression possible for 
these quantities consistent with the condition that the forces between 
closed circuits should agree with those given by Ampére’s laws, 
Paz —y e+e [la dn dé, 
Let us put 
with similar expressions for G and H, where & is a constant and 
dr dr dr ; 
¥= |([n( u get v Ty + w Zz) dé dn dé. 
Transforming this expression we see, using the same notation as before, 
that 
Y= || re {0 Gr —ua)-+m (1-09) +n (wo —m)} dS 
du , dv , dw - 
ids ges EAN Ee) 
Nal ee gee 
=| [ur Bds— il} pr P dé dn dé, 
where dS is an element of a surface at which there is discontinuity in: 
U, 0, W. 
Let us now consider the equations which hold in a perfectly insulating 
dielectric. 
The rate of change of the z component of the electromotive force in a 
medium at rest 
a4 a?k de d*p 
a dt das 
where ¢ is the electrostatic potential ; it also equals f/n, so that 
