Conway—Llectro-mugnetie Mass. 53 
tangent plane be denoted by the single accent, and on the negative side by the 
double accent, and let w be the normal velocity of the surface, then :— 
— wV?(X’ — X”") + 4, = — (B'— f’), 
= 0 V3 (Y’— VY") + 4ni, = a! — a”, 
— wV?(Z' — Z") + Aris = 0, 
wa =a?) == (Y= ¥), 
w (B'— B") = X"— X", 
yoy Su. 
UI.—Tse FunpAMENTAL ASSUMPTION. 
In dynamics we pass from the case of a particle to that of a rigid body 
by means of D’Alembert’s principle. 
This principle, by asserting that the internal reactions of such a body are 
in equilibrium amongst themselves, enables us to dispense with such reactions 
in our equations. The assumption which we are going to make use of here 
is of a similar nature, and is as follows:—The internal electro-magnetic forces of an 
electrical system are in equilibrium with the external electro-magnetic forces. 
ITV.—Denpuctions or Expressions ror Momentum AND ENERGY. 
Let us consider an electrical system of density p, the current vector being 
(zi, %, %), and let the forces arising from this be (X, Y, Z) and (a, B, y), the 
external field of force in which the system is situated being denoted by 
(Xo, Yo, 4) (a, By y)- The internal force along the z-axis acting on the 
element dz is (Xp + yt: — $i), dr; and the external force is X,p + y,l2 — Byts. As 
a first example let us consider the activity of the forces. Taken collectively 
they are in equilibrium ; hence the total activity is zero, or the reversed activity 
of the internal forces is the rate at which energy is supplied from outside, 7.¢., the 
former expression gives us the rate of change of the electro-magnetic energy 
of the system. The reversed activity is obviously 
— fffdr (Xa + Yi, + ZH). 
Substituting for (4, la, #) from the circuital relations, we find that the subject 
of integration becomes 
(Sr) V*/0t(X? + Y? + Z*) + (Sr) 0/ot(a? + B+ y’) 
= (4a) 10/ax [BZ — yY] + dley (yX — aZ) + d/ez(a¥ — BX). 
