Relation of Mass to Energy. 5 
For the total density of energy-flow (fx,fy,fz) we must 
of course add to the above the components of the Poynting 
vector. Writing as usual X, Y, Z and ec, fi % 7 for the electric 
and magnetic force intensities and calling (V) the velocity of 
light, we have 
U = J (r* -/3Z) - («bX, + *y. + r.-z, J, ■ 
V I 
/» = 4^. O z - 7X) - (r,X, + r y Y y + c-Z y ), > . . (2) 
/l- = ~(j3X - *Y) - (r,X, + v s Y 2 + r,Z 2 ), j 
These equations give the density of the total energy-flow 
through any purely electrical system, in which the ordinary 
electrical laws hold universally/ 
6. Consider an isolated electrical system moving as a whole 
through space with the constant velocity («i). A constant 
velocity will be possible if the system retains on the average 
the same internal structure. The total average rate of transfer 
of energy corresponding to the movement of such a system 
is evidently (r x . W), where W is the total contained energy. 
Another expression for the same thing is to be obtained by 
integrating throughout the system the components along (i\) 
°£ (/>•> fy-> fz) given in equations (2). In order that the 
velocity (r 2 ) may appear explicitly, however, it is necessary 
that the velocity (v), which was used in equations (2), be 
written as the sum of (v{) and another velocity (v 2 )- Then 
(v 2 ) is the velocity with respect to axes moving with the 
system. 
If 1, m, n are the direction cosines of the constant velocity 
(rj), we have for the total energy-flow (F) in the direction 
6 'i = F = j (lf x + mf y + nf-)dr 
+ -^ \ (aZ-^yX)dT—mv 1 1 (llLy+mYy + nZ^dT— m j (v 2 xXg+V2i,Y 3 , + V2zZy)dT 
+ |^ j 08X-«Y)rfT-.n», [(IXz + mYe+nZ^dT-n \(v 2X X,-{ v 2y Y z + m z Z z )di 
.... (3) 
