Page:Lorentz Simplified1899.djvu/8

 Dividing this by (3), we get

$$\frac{\mathfrak{p}_{x}}{V}\cdot\frac{a}{VT}$$,

an extremely small quantity, because the diameter of the ions is a very small fraction of the wave-length. This is the reason why we may omit the last term in (Ic).

As to the equations (Vc), it must be remarked that, if the displacements are infinitely small, the same will be true of the velocities and, in general, of all quantities which do not exist as long as the system is at rest and are entirely produced by the motion. Such are $$\mathfrak{H}'_{x},\mathfrak{H}'_{y},\mathfrak{H}'_{z}$$. We may therefore omit the last terms in (Vc), as being of the second order.

The same reasoning would apply to the terms containing $$\frac{\mathfrak{p}_{x}}{V^{2}}$$, if we could be sure that in the state of equilibrium there are no electric forces at all. If, however, in the absence of any vibrations, the vector $$\mathfrak{F}'$$ has already a certain value $$\mathfrak{F}_{0}'$$, it will only be the difference $$\mathfrak{F}'-\mathfrak{F}_{0}'$$, that may be called infinitely small; it will then be permitted to replace $$\mathfrak{F}'_{y}$$ and $$\mathfrak{F}'_{z}$$ by $$\mathfrak{F}'_{0y}$$ and $$\mathfrak{F}'_{0y}$$.

Another restriction consists in supposing that an ion is incapable of any motion but a translation as a whole, and that, in the position of equilibrium, though its parts may be acted on by electric forces, as has just been said, yet the whole ion does not experience a resultant electric force. Then, if $$d\tau$$ is an element of volume, and the integrations are extended all over the ion,

Again, in the case of vibrations, the equations (Vc) will only serve to calculate the resultant force acting on an ion. In tho direction of the axis of y e. g. this force will be

$$\int\varrho\mathfrak{E}_{y}d\tau$$.

Its value may be found, if we begin by applying the second of the three equations to each point of the ion, always for the same universal time t, and then integrate. From the second term on the right-hand side we find