108.

Let us proceed in a perfectly general way to consider the components of the field-strengths $⋿$ and $𝖧$ as functions of the time at a definite point, which we may think of as the origin of the coordinate system. Of these components, which are produced by all rays passing through the origin, there are six; we select one of them, say ${⋿}_z$, for closer consideration. However

complicated it may be, it may under all circumstances be written as a Fourier's series for a limited time interval, say from $t=0$ to $t=𝖳$; thus ${⋿}_z={\sum }_{n=1}^{n=\infty }{C}_n\cos (\frac{2\pi nt}{𝖳}-{\theta }_n)\mathrm{\backslash Label}[eqn](149)\text{upshape (149)}$ where the summation is to extend over all positive integers $n$, while the constants $C_n$ (positive) and ${\theta }_n$ may vary arbitrarily from term to term. The time interval $𝖳$, the fundamental period of the Fourier's series, we shall choose so large that all times $t$ which we shall consider hereafter are included in this time interval, so that $0<t<𝖳$. Then we may regard ${⋿}_z$ as identical in all respects with the Fourier's series, i.e., we may regard ${⋿}_z$ as consisting of "partial vibrations," which are strictly periodic and of frequencies given by $\nu =\frac{n}{𝖳}.$

Since, according to [sect:3.] Sec. 3, the time differential $dt$ required for the definition of the intensity of a heat ray is necessarily large compared with the periods of vibration of all colors contained in the ray, a single time differential $dt$ contains a large number of vibrations, i.e., the product $\nu dt$ is a large number. Then it follows a fortiori that $\nu t$ and, still more, $\nu 𝖳=n\text{ is enormously large}\mathrm{\backslash Label}[eqn](150)\text{upshape (150)}$ for all values of $\nu$ entering into consideration. From this we must conclude that all amplitudes $C_n$ with a moderately large value for the ordinal number $n$ do not appear at all in the Fourier's series, that is to say, they are negligibly small.