250 
^IR. J. LARMOR A DYNAMICAL THEORY OF 
by integration by parts to reduce the magnitude of the c[uantity that remains under 
the sign of volume integration, so that in the limit we may be able to neglect that 
part; thus we obtain |Xh/r = | PdS + J (p'+ p) Pdr. By the defin¬ 
ition of electric force, (P, Q, R) is the force due to a volume distribution of density 
p-\- p and to extraneous causes ; so that in the limit when the transitional layer is inde¬ 
finitely thin, we have, by Coulomb’s principle, J {p + p) Pc/r = |-1 (cr'+ o') (R 2 + Pi) 
= ( 877 )“^ KNs—Ni) (P 2 +P 1 ) tZS, Pi, P .3 being the values of the a? component P, and Nj, 
Ng those of the normal component N, of the electric force (P, Q, R) on the two sides of 
the layer, all measured towards the side 2 , while o-' and cr are the surface densities consti¬ 
tuted in the limit by the aggregates of p and p respectively taken throughout the layer. 
Hence in the limit + PfZS| -f-( 87 r)“^|(N 3 — Ni) (Pq- p Pi) dS. 
Thus the electric traction on the interface of transition may be represented by a 
pull towards each side, along the direction of the resultant electric force F ; this pull 
is on the side 2 of intensity n — -J {n\ — n\ — a) Fg, that is ^ (cr -f- 3 -f- n 1 ) F 3 in 
the direction of Fg, where n' is the normal component of the polarization of the medium 
measured positive towards the side 2 ; on the face 1 the pull is 4 (cr — 3 — ^i-^Fi 
now in the direction of Fi, n' being measured positive as before. As the tangential 
component of the electric force F is under all circumstances continuous across the 
interface, the total traction on both sides combined is along the normal, and equi¬ 
valent to ^ (n'g + n\) (Ng — Ni) together with the tractions ^crFg, ^crFi acting on 
the true charge cr, all the quantities being now measured positive in any the same 
direction. If n" denote the normal component of the total displacement (/", g", h"), 
so that 7 i" = N/Itt + n', n', — = cr, the first part of this total traction is 
1 ( 71''3 + n'\ — No/Itt — Nj/Itt) (Ng — Nj), -which is simply — + 2 Trn\~ towards 
the side 2 .* When the interface is between a dielectric 1 and a conductor 2 , the 
traction is only towards the side 1 and is equal to ^{n'l 4 ’ cr) Fj, or |- 7 i\Fj, per 
unit area, along the normal which is now the direction of the resultant force. 
All this is quite independent of the law of the connexion between the polarization 
and the electric force in the material medium. Thus, under the must general 
circumstances as regards electric field, whether there is material equilibrium or not, 
the forcive on the material due to its electric excitation consists of the interfacial 
tractions thus specified, together with a force (X', Y', Z') and a torque (L', M, N ) per 
unit volume, given by the fonnulm (X', Y', Z') = [diclx, djdy, djdz) (/'P 4-Q 4"^^ I^) 
and (L', M', N') = (^'R-/ 7 'Q, /fP-/Ti, /Q-^T), in the former of which (/, cj, li') 
is not to be ditferentiated. 
The assumption underlying this analysis, that the transitions are gradual, will be 
* It luiiy be recalled that in the terminology here employed, the true electrification <t is the density of 
unpaired electrons: while the true electric current arises from the movements of all the electrons, free 
and paired, but does not include the change of mthereal strain which must be added in order to make 
up the total cireuital current of Maxwell. 
