226 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
under the influence of a powerful extraneous magnetic field ; when the magnetic 
field is due solely to its own motion the redistribution is of course absolutely 
negligible. 
(ii.) In the case of a dielectric (as also in the above) the restriction to a steady 
state and permanent configuration may be dispensed with ; for the magnetic field 
arising from induced displacement currents can. always be neglected in comparison 
with the inducing field. Thus, {a, h, c) being the extraneous inducing field, the 
electric forces inside and outside a rotating mass are 
{o)CX — cbV-^/dx, cocy — d^Jdy, — wax — why — d'^Vjdz) and — {djdx, djdy, djdz) 
As there can be no free electrification, 
V^'Fj = (I — K“^) 6 j { 2 c -b x{dcldx — dajdz) + y{dcldy — dhjdz)] and = 0 ; 
while at the surface 
= ■^ 2 ) and K d'¥^jdn — (K — 1 ) cu {cxl + cym — [ax + hy) n) = d'^Jdn, 
the outside medium being air. If the dielectric body is a sphere rotating in a 
uniform field ( 0 , 0 , c) parallel to the axis, this gives by the usual harmonic anaWsis 
= A (1 — K“^) wcr~ + A)' cos A' and ^3 = cos d-j- where, 7 ’^ being the 
radius, A = B/r^^^ - 3K/(2K+ l)ri. A'= - |/(K+ 2 ) B'=(K~ 1 );'(K+ 2 ). o^cr^; 
thus determining the electric potential Tg in the space surrounding the rotating 
sphere. 
15. More generally, let us consider steady distributions of electric charges on a 
system of conductors and dielectric bodies in motion through the aether. That there 
may be a steady state, without conduction currents, it is necessary that the configu¬ 
ration of the matter shall be permanent, and that its motion shall be the same at all 
times relative to this configuration and to the aether, and also to the extraneous 
magnetic field if there is one : this confines it to uniform spiral motion on a definite 
axis fixed in the aether. Referring to axes fixed in the material system, the vector 
potential has in the steady motion no time-variation : hence 
(P, Q, R)= —{djdx, djdy, c//rfo)V, (P',Q', R') = (P — <70 -f ?' 6 ,Q — ra 4 -^c,R — ^76 -f qa). 
The magnetic induction through any circuit moving with the matter being constant, 
(P, Q, R) is derived (§ 12 ) from an electric potential function V. Inside a conductor 
the electric force must vanish, otherwise electric separation woidd be going on, 
therefore V must there be constant. 
When the surrounding dielectric is free space, the total current in it, referred to 
these axes moving with the matter, is — (pdjdx + qdjdy -f rdjdz) [f,g, h). When 
the velocity [p, g, r) of the matter is uniform, it then follows from A:m]’ERe’s 
circuital relation that [a, h, c) = in [qh — rg, rf — pli, pg — qf). Hence (/', g, h), 
given by = P — Q'C + rh, is expressed in terms of (P, Q, R) by equations of 
t}’pe (c" — qr' — q~ — R^)./ = P/Itt — (pP + g'Q + 7’R). The circuital quality 
of [f, g, h) thus gives the characteristic equation of the single independent variable 
