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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 Poynting's theorem by

$\int _{t_{0}-T}^{t_{0}}dt\int [{\mathfrak {d.H}}]_{n}d\sigma =0$

(105)

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

$\int _{t_{0}-T}^{t_{0}}dt\int [{\mathfrak {d}}'.{\mathfrak {H}}']_{n}d\sigma =0$

(106)

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

$V^{2}[{\mathfrak {p.H}}]_{n}$,

or, as we find from the formulas (IX) and ($VI_{b}$) (§§ 56 and 20),