Page:Electromagnetic phenomena.djvu/23

in the state of rest, and by U+U' the energy in the state of motion, we have by the formulae of this paper, up to the terms of the second order,

$U'={\frac {2w^{2}}{c^{2}}}U$,

an expression, agreeing in order of magnitude with the value used by Trouton for estimating the effect.

The intensity of the sudden jerk or impulse will therefore be ${\frac {U'}{w}}$.

Now, supposing the apparatus to be initially at rest, we may compare the deflexion α, produced by this impulse, to the deflexion α' which may be given to the torsion-balance by means of a constant couple K, acting during half the vibration time. We may also consider the case in which a swinging motion has already been set up: then the impulse, applied at the moment in which the apparatus passes through the position of equilibrium, will alter the amplitude by a certain amount β and a similar effect β' may be caused by letting the couple K act during the swing from one extreme position to the other. Let T be the period of swinging and l the distance from the condenser to the thread of the torsion-balance. Then it is easily found that

${\frac {\alpha }{\alpha '}}={\frac {\beta }{\beta '}}={\frac {\pi U'l}{KTw}}$.

(39)

According to Trouton's statements U' amounted to one or two ergs, and the smallest couple by which a sensible deflexion could be produced was estimated at 7,5 C. G. S.-units. If we substitute this value for K and take into account that the velocity of the Earth's motion is 3 × 10⁶ c.M. per sec., we immediately see that (39) must have been a very small fraction.