524 
PROFESSOR H. LA^IB ON ELECTRICAL 
where, for the moment, the coordinates x, y, z refer to the conductors, and £, y, £ to 
the dielectric. Let us form the equation of energy for the case where disturbances 
produced anyhow in the field are left to themselves. 
We have 
^7 1 p m ^ 
if j | f'u-\-¥u-\-kc.)dxdyclz 
+ i|| (Ff+~Ff+&c.)dgdr)dt 
= [) (lb/ + Gv-\-lS.w)dxdydz 
+ [[{(tf+&g+H.h)d£dydi • • 
Substituting the values of F, G, H from (3) and (4), we find 
-= — [| \^p{u^ fv 2 fw^jdxdydz 
cfT 
dt 
— 4 7t r' I 
’+gg+hh)dgdyd£ 
f : + f +l f) dxd y dz 
(15)' 
Idie last two integrals disappear in virtue of the solenoidal conditions satisfied by 
the flow of electricity.! Hence 
|(T+w)=-f[■ ■ 
(16). 
This expresses that the electrical energy lost is equivalent to the heat generated in 
the conductors according to Joule’s law. 
2. Now let us suppose that F, G, H, &c., all vary as e M . The electrical motions in 
the conductors and in the surrounding dielectric may be of two kinds, free and forced. 
In the various modes of free motion the corresponding values of X are real and 
negative. In the case of forced motion the disturbing force at any point of the 
field may, by Fourier’s (double-integral) theorem, be expanded, as regards the time, 
in a series of periodic terms. The effects of these can then be investigated separately 
and afterwards superposed. The value of X corresponding to any one term is X=27rty>, 
where p is the frequency, and i— f — 1. 
* This may be deduced from (1) by Green’s Theorem. It is a particular case of Thomson and Tait, 
§ 313 (/). 
f Maxwell’s ‘ Electricity,’ § 100a. 
