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 For the total density of energy-flow ($$f_{x}, f_{y}, f_{z}$$) we must of course add to the above the components of the Poynting vector. Writing as usual $$X, Y, Z$$ and $$\alpha, \beta, \gamma$$ for the electric and magnetic force intensities and calling ($$V$$) the velocity of light, we have

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 ($$v_{1}$$). 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 ($$v_{1} \cdot W$$), 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 ($$v_{1})$$ of $$(f_{x}, f_{y}, f_{z}$$) given in equations (2). In order that the velocity ($$v_{1}$$) may appear explicitly, however, it is necessary that the velocity ($$v$$), which was used in equations (2), be written as the sum of ($$v_{1}$$) and another velocity ($$v_{2}$$). Then ($$v_{2}$$) is the velocity with respect to axes moving with the system.

If $$l, m, n$$ are the direction cosines of the constant velocity ($$v_{1}$$), we have for the total energy-flow ($$F$$) in the direction of ($$v_{1}$$),