222 
ME. J. LARMOR ON A DYNAMICAL THEORY OF 
(P, Q, R) denote the electric force, namely that which acts on the electrons, and 
(P', Q', R') the sethereal force, that which produces the eethereal electric displace¬ 
ment [f. g, h ); let p denote density of free electric charge. Then the electromotive 
equations are* 
wliere 
and 
where 
qc — rb -—-“ 
^ dt dx 
F 
f * + I(b i 
p, _ _ 
dt dx 
C — f/r, a 
dyj r 
f——~ P' 
d TttC”- ’ 
dH _ . 
dy dz 
“ = f + f + 
d(/+/) . d{g' + cj) . d(/d+A) 
P = —y:— + —z:— + 
da; 
dy 
dz 
From the formula for (P, Q, R) Faraday’s law follows that the line integral of electric 
force round a circuit in uniform motion ivitli the matter is equal to the time-rate of 
diminution of the magnetic flux through its aperture. The line-integral of the 
^ This scheme forms an improved summary of that worked out in Part II. §§ 15-19; the expressions 
there assigned for p and ^ have here been corrected, and Vq, Wq') is merged. 
t [Added 8ept. 14.—The term 6f'jdt in 7i arises as follows. In addition to the change of the polariza¬ 
tion in the element of volume, dfjdt, there is the electrodynamic effect of the motion of the positive 
and negative electrons of the polar molecule. Now the movement of two connected positive and 
negative electrons is equivalent to that of a single jjositive electron round the circuit formed by 
joining together the ends of their paths: and a similar statement holds when there are more than two 
electrons in the molecule. Hence the motion of a polarized medium with velocity (^, g, 7 -), which 
need not be constant from point to point, produces the electrodynamic effect of a magnetization 
{rg' — qh\ ph' — rf, qf — pg') distributed throiighout the volume: cf. Part I, § 125. And it has been 
shown in Part II, § 31 that any distribution of magmetism (A, B, C) may be represented as a volume 
distribution of electric current equal to curl (A, B, C), which is necessarily circuital, togetlier with a 
surface current sheet equal to (B?z— Cm, Cl — Aw, A7)i — BZ). Thus, ivhen (y), q, r) is uniform 
and (/', g, 72) circuital, the above magnetic distribution is equivalent to a current S 3 'stem 
(pdjdx + qdjdy + rdjdz) {f,g', 72) together with current sheets on interfaces of discontinuity: this 
system is to be added on to djdt (/', g', h') in order to give the full electrodjmamic effect. Thus in 
these special circumstances the formulation in the test is correct in so far as it leads to the correct 
diferential equations for the element of the medium: the integral expression there given forP is how¬ 
ever only correct either when it is reduced to the differential form — v-F/Itt — u dCjdy — dBldz, 
which is derivable on integration of its second terra by parts, or else when, the velocity of the matter 
still being uniform, discontinuous interfaces are replaced in the analysis by gradual though rapid tran¬ 
sitions. These conditions are satisfied in all the apiplications that follow : but they would not be satisfied 
for example in the problem of the reflexion of radiation from the surface of moving matter. 
But a formulation which is pmeferable to the above, in that it is absolutely general, is simply to im¬ 
plicitly include the above virtual magnetization directly in (A, B, C) and consequently change from 
rf jdt to df jdt in the expression for u : this wall also involve that the relation A = kx which occurs lower 
down shall be replaced by A = /ira -p rg' — qli', but there will be no further alteration in the argument 
of the text.. The boundarq^ conditions of the text ai’e unaltered.] 
