174.

The distinction between rapidly variable and slowly variable quantities introduced in the preceding section has, at the present stage, an important physical aspect, because in the following we shall assume that only slow variability with time is capable of direct measurement. On this assumption we approach conditions as they actually exist in optics and heat radiation. Our problem will then be to establish relations between slowly variable quantities exclusively; for these only can be compared with the results of experience. Hence we shall now determine the most important one of the slowly variable quantities to be considered here, namely, the "spectral intensity" $𝖨$ of the exciting vibration. This is effected as in [eqn:(158)] (158) by means of the equation $J={\int }_0^{\infty }𝖨d\nu .$

By comparison with [eqn:(313)] (313) we obtain: $\left{

$$\begin{gathered} 𝖨=\int d\mu ({𝖠}_{\mu }\cos 2\pi \mu t+{𝖡}_{\mu }\sin 2\pi \mu t) \\ \text{whereqquad}[t]{𝖠}_{\mu }\&=\overline{C_{\nu +\mu }{C}_{\nu }\cos ({\theta }_{\nu +\mu }-{\theta }_{\nu })\mathrm{\backslash Strut}} \\ {𝖡}_{\mu }\&=\overline{C_{\nu +\mu }{C}_{\nu }\sin ({\theta }_{\nu +\mu }-{\theta }_{\nu })\mathrm{\backslash Strut}}. \end{gathered}$$

\right. \Label[eqn]{(316)}\tag*{\upshape (316)}$

By this expression the spectral intensity, $𝖨$, of the exciting vibration at a point in the spectrum is expressed as a slowly variable function of the time $t$ in the form of a Fourier's integral. The dashes over the expressions on the right side denote the mean values extended over a narrow spectral range for a given value of $\mu$. If such mean values do not exist, there is no definite spectral intensity.