Page:Lorentz Simplified1899.djvu/9

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( 435 )

,

or, since we may replace by and by ,

,

which vanishes on account of (5).

Hence, as far as regards the resultant force, we may put , that is to say, we may take as the electric force, acting not only on ions at rest, but also on moving ions.

The equations will be somewhat simplified, if, instead of , we introduce the already mentioned difference . In order to do this, we have only twice to write down the equations (Ic)—(IVc), once for the vibrating system and a second time for the same system in a state of rest; and then to subtract the equations of the second system from those of the first. In the resulting equations, I shall, for the sake of brevity, write instead of , so that henceforth will denote not the total electric force, but only the part of it that is due to the vibrations. At the same time we shall replace the value of , given above, by

.

We may do so, because we have supposed to have the same values all over an ion, and because is independent of the time, so that

.

Finally we have

, (Id)
, (IId)
, etc. (IIId)
, etc. (IVd)