ON ELECTRICAL THEORIES. 129 
Maxwell’s assumption is that a=K/4, and this makes the equations 
much simpler; it is, however, important to remember that Maxwell’s 
theory of the dielectric involves the two assumptions— 
Ist. That alterations in the dielectric polarisation produce effects 
analogous to those of ordinary conduction currents ; 
2nd. That the magnitude of the equivalent conducting current 
=a { = B} /dt, where F is the electromotive force at the point ; this is 
Tv 
equivalent to saying that all the currents are closed currents, and that 
there is no discontinuity in them. 
Maxwell developes his theory by means of the principle of the Con- 
servation of Energy. 
Let us consider an electric field full of currents, whether ordinary 
conduction currents or polarisation ones. Then this field may be looked 
upon as a material system, and all the phenomena have to be explained as 
the effects of the motion of this system ; a current must be looked upon 
as a change in the structure of the system, and so capable of representa- 
tion by means of the differential coefficients of the co-ordinates fixing 
the system; we can thus represent the current at each point as the 
differential coefficient of some generalised co-orilinate fixing the system ; 
the components uw, v, w of the current passing through an element dz, 
dy, dz may be looked upon as the rates of change of some generalised 
co-ordinates ; we may write the energy as 
a(|{ cw + Gv + Hw} de dy da, 
where F', G, H may be looked upon as momenta corresponding to 
u,v, w. It remains to identify F, G, H with known quantities. Maxwell 
does this by the aid of Faraday’s result, that the electromotive force 
round a circuit equals the rate of diminution of the number of lines of 
force passing through it. 
Let us consider a single linear circuit in which the current is 7, or 
say dq/dt, then the energy 
dq { dz dy et 
=}/2 = at -s 
| Ee eee + 
where ds is an element of circuit; but by Lagrange’s equation the force 
tending to increase q, i.e., the electromotive force in the circuit, 
d da dy dz 
=— |. Pye —~ + H— jds; 
zi ll tik z) o3 
so that (F ioe Sere Seed +HS)as 
ds ds ds 
equals the number of lines of force passing through the circuit ; but if dS 
be an element of surface closing up the circuit, 7, m, n the direction cosines 
of the normal, then by Stokes’ theorem f 
daz dy dz 
\(®. a +G abe HS ) ds 
dH dG dF dH aqaG@ dF 
(hi dhe aeRirese a See SS ; 
ih ‘a = )t™ dz mt da 7) f 483 
1885. K. 
