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The conservation of energy in a more general case.
§ 79. An arbitrary transparent body K shall be hit by a homogeneous light motion, whose intensity remains constant; consequently, a certain motion arises in the body and in the aether in its vicinity.

Here, when Earth is at first imagined as stationary, the components of $$\mathfrak{d}$$ and $$\mathfrak{H}$$ in the aether are certain functions of x,y,z,t, and namely as regards the last variable, goniometric functions with the period T. During a complete period, e.g. in the time interval from $$t_{0}-T$$ to $$t_0$$, equal quantities of energy must flow in- and outwards through an arbitrary surface σ that surrounds the surface, which can be expressed by 's theorem by

By assuming, that this condition is fulfilled, we want to show, that also state of motion that corresponds with that above, which can exist in the case of a translation $$\mathfrak{p}$$, satisfies the energy theorem.

If we replace in the functions, which apply to $$\mathfrak{d}_{x},\ \mathfrak{H}_{x}$$, etc. when the Earth is at rest, the time t by the "local time" $$t'$$ (§ 31), and if we understand in those functions by x, y, z the coordinates with respect to a movable system, then we obtain values of $$\mathfrak{d}'_{x},\ \mathfrak{H}'_{x}$$, etc. for the new state. From (105) it thus directly follows, that

if it is presupposed, that we choose for σ a surface, which shares the motion of the body.

§ 80. However, now the flux of energy through a fixed surface $$\sigma$$ shall now be considered. The energy flux related to its unit shall be

or, as we find from the formulas (IX) and ($$VI_b$$) (§§ 56 and 20),