148.

Let us consider an oscillator which has just completed an emission and which has, accordingly, lost all its energy of vibration. If we reckon the time $t$ from this instant then for $t=0$ we have $f=0$ and ${\displaystyle \frac{df}{dt}}=0$, and the vibration takes place according to equation [eqn:(233)] (233). Let us write ${⋿}_z$ as in [eqn:(149)] (149) in the form of a Fourier's series: ${⋿}_z={\sum }_{n=1}^{n=\infty }[A_n\cos \frac{2\pi nt}{𝖳}+B_n\sin \frac{2\pi nt}{𝖳}],\mathrm{\backslash Label}[eqn](235)\text{upshape (235)}$ where $𝖳$ may be chosen very large, so that for all times $t$ considered $t<𝖳$. Since we assume the radiation to be stationary, the constant coefficients $A_n$ and $B_n$ depend on the ordinal numbers $n$ in a wholly irregular way, according to the hypothesis of natural radiation ([sect:117.] Sec. 117). The partial vibration with the ordinal number $n$ has the frequency $\nu$, where $\omega =2\pi \nu =\frac{2\pi n}{𝖳},\mathrm{\backslash Label}[eqn](236)\text{upshape (236)}$ while for the frequency ${\nu }_0$ of the natural period of the oscillator ${\omega }_0=2\pi {\nu }_0=\sqrt{\frac{K}{L}}.$

Taking the initial condition into account, we now obtain as the solution of the differential equation [eqn:(233)] (233) the expression $f={\sum }_1^{\infty }[a_n(\cos \omega t-\cos {\omega }_{0}t)+b_n(\sin \omega t-\frac{\omega }{{\omega }_0}\sin {\omega }_{0}t)],\mathrm{\backslash Label}[eqn](237)\text{upshape (237)}$ where $a_n=\frac{A_n}{L({\omega }_0^2-{\omega }^2)},b_n=\frac{B_n}{L({\omega }_0^2-{\omega }^2)}.\mathrm{\backslash Label}[eqn](238)\text{upshape (238)}$

This represents the vibration of the oscillator up to the instant when the next emission occurs.

The coefficients $a_n$ and $b_n$ attain their largest values when $\omega$ is nearly equal to ${\omega }_0$. (The case $\omega ={\omega }_0$ may be excluded by assuming at the outset that ${\nu }_0𝖳$ is not an integer.)