THE ELECTRIC AHD LUMINIFEROUS MEDIUM. 
759 
same across all sections of the wire, and that the energy relations are expressed 
in the same manner as before in terms of the total current. 
The electrodynamic forces between linear current-systems are thus fully involved 
in the kinetic-energy function of the sethereal medium. The only point into which 
we cannot at present penetrate is the precise nature of the surface-action by which 
the energy is transferred (just as in § 45) from the electric medium to the matter of 
the perfect conductor; all the forces of the field are in fact derived from their appro¬ 
priate energy-functions, so that it is not necessary, though it is desirable, to know 
the details of the interaction between rnther and matter, at the surface of a con¬ 
ductor. 
Mathematical Analysis of Electro-Kinetic Forces and their reaction on the Material 
Medium. 
54. We have shown that the electro-kinetic energy of a system of linear electric 
currents may be expressed in the form 
t 
47rT = 11^ ds^ ds, + 11^ ds^ ds,, 
the velocity-system which they involve being sufficiently described by the set of 
velocity co-ordinates • combined with the kinetic constraints derived from 
the constitution of the mther. To mark that these quantities are dynamically 
velocities, let us denote tj, . . . by dejdt, defdt, ... so that e^, Cg,. , . will be taken 
as electric co-ordinates of position. The general variational equation of motion may 
be expressed in the form 
8 |t = |8 Wi dt + |(Ei 8ei + Eg 8^2 fi- . . .) dt, 
where E^ is by definition such that E^ 8 ^^ is the work done in the system during a 
displacement 8 ep so that in electric phraseology E^ with sign changed is the electric 
force integrated round the circuit 1 , or the electromotive force in that circuit. Also 
Wj is any other potential energy the system may possess ; the energy of electric strain 
throughout the medium being now very small, as there are no static electrifications, 
and the motions are supposed slow compared with the velocity of radiation. Thus, 
adopting the notation of coefficients of electrodynamic induction, so that 
d(f 
de,-^ de„ 
dt dt 
+ 
12 ) 
depending on the configurations of the circuits, we have 
de^ . \ d Scj 
dt 
dt 
+ |8iT dt, 
