173.

The conception of the intensity of vibration $J$ necessarily contains the assumption that this quantity varies much more slowly with the time $t$ than the vibration ${⋿}_z$ itself. The same follows from the calculation of $J$ in the preceding paragraph. For there, according to [eqn:(312)] (312), $\nu \tau$ and ${\nu }^{\prime }\tau$ are large, but $({\nu }^{\prime }-\nu )\tau$ is small for all pairs of values $C_{\nu }$ and $C_{{\nu }^{\prime }}$ that come into consideration; hence, a fortiori, $\frac{{\nu }^{\prime }-\nu }{\nu }=\frac{\mu }{\nu }\text{is small},\mathrm{\backslash Label}[eqn](315)\text{upshape (315)}$ and accordingly the Fourier's integrals ${⋿}_z$ in [eqn:(311)] (311) and $J$ in [eqn:(314)] (314) vary with the time in entirely different ways. Hence in the following we shall have to distinguish, as regards dependence on time, two kinds of quantities, which vary in different ways: Rapidly varying quantities, as ${⋿}_z$, and slowly varying quantities as $J$ and $𝖨$ the spectral intensity of the exciting vibration, whose value we shall calculate in the next paragraph. Nevertheless this difference in the variability with respect to time of the quantities

named is only relative, since the absolute value of the differential coefficient of $J$ with respect to time depends on the value of the unit of time and may, by a suitable choice of this unit, be made as large as we please. It is, therefore, not proper to speak of $J(t)$ simply as a slowly varying function of $t$. If, in the following, we nevertheless employ this mode of expression for the sake of brevity, it will always be in the relative sense, namely, with respect to the different behavior of the function ${⋿}_z(t)$.

On the other hand, as regards the dependence of the phase constant ${\theta }_{\nu }$ on its index $\nu$ it necessarily possesses the property of rapid variability in the absolute sense. For, although $\mu$ is small compared with $\nu$, nevertheless the difference ${\theta }_{\nu +\mu }-{\theta }_{\nu }$ is in general not small, for if it were, the quantities $A_{\mu }$ and $B_{\mu }$ in [eqn:(314)] (314) would have too special values and hence it follows that ${\displaystyle \frac{\partial {\theta }_{\nu }}{\partial \nu }}·\nu$ must be large. This is not essentially modified by changing the unit of time or by shifting the origin of time.

Hence the rapid variability of the quantities ${\theta }_{\nu }$ and also $C_{\nu }$ with $\nu$ is, in the absolute sense, a necessary condition for the existence of a definite intensity of vibration $J$, or, in other words, for the possibility of dividing the quantities depending on the time into those which vary rapidly and those which vary slowly–-a distinction which is also made in other physical theories and upon which all the following investigations are based.