135.

being loaded, say at its positive pole, with a comparatively large inert mass, which is perfectly neutral electrodynamically, in order to decrease its velocity for a given kinetic energy below any preassigned value whatever. Of course this consideration remains valid also, if, as is now frequently done, all inertia is reduced to electrodynamic action. For this action is at any rate of a kind quite different from the one to be considered in the following, and hence cannot influence it.

Let the state of such an oscillator be completely determined by its moment $f(t)$, that is, by the product of the electric charge

of the pole situated on the positive side of the axis and the pole distance, and by the derivative of $f$ with respect to the time or $\frac{df(t)}{dt}=\dot{f}(t).\mathrm{\backslash Label}[eqn](204)\text{upshape (204)}$ Let the energy of the oscillator be of the following simple form: $U=\frac{1}{2}K{f}^2+\frac{1}{2}L{\dot{f}}^2,\mathrm{\backslash Label}[eqn](205)\text{upshape (205)}$ where $K$ and $L$ denote positive constants, which depend on the nature of the oscillator in some way that need not be discussed at this point.

If during its vibration an oscillator neither absorbed nor emitted any energy, its energy of vibration, $U$, would remain constant, and we would have: $dU=Kfdf+L\dot{f}d\dot{f}=0,\mathrm{\backslash Label}[eqn](205a)\text{upshape (205a)}$ or, on account of [eqn:(204)] (204), $Kf(t)+L\ddot{f}(t)=0.\mathrm{\backslash Label}[eqn](206)\text{upshape (206)}$ The general solution of this differential equation is found to be a purely periodical vibration: $f=C\cos (2\pi \nu t-\theta )\mathrm{\backslash Label}[eqn](207)\text{upshape (207)}$ where $C$ and $\theta$ denote the integration constants and $\nu$ the number of vibrations per unit time: $\nu =\frac{1}{2\pi }\sqrt{\frac{K}{L}}.\mathrm{\backslash Label}[eqn](208)\text{upshape (208)}$