Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/373

 

second series of transits, and deduce the time of vibration $T^{\prime }$ and the time of middle transit $P^{\prime }$, noting the direction of this transit.

If $T$ and $T^{\prime }$ the periods of vibration as deduced from the two sets of observations, are nearly equal, we may proceed to a more accurate determination of the period by combining the two series of observations.

Dividing $P^{\prime }-P$ by $T$, the quotient ought to be very nearly an integer, even or odd according as the transits $P$ and $P^{\prime }$ are in the same or in opposite directions. If this is not the case, the series of observations is worthless, but if the result is very nearly a whole number $n$, we divide $P^{\prime }-P$ by $n$, and thus find the mean value of $T$ for the whole time of swinging.

740.] The time of vibration $T$ thus found is the actual mean time of vibration, and is subject to corrections if we wish to deduce from it the time of vibration in infinitely small arcs and without damping.

To reduce the observed time to the time in infinitely small arcs, we observe that the time of a vibration of amplitude $\alpha$ is in general of the form

where $\kappa$ is a coefficient, which, in the case of the ordinary pendulum, is ${\frac {1}{64}}$. Now the amplitudes of the successive vibrations are $c$, $c\rho ^{-1}$, $c\rho ^{-2}$, … $c\rho ^{1-n}$, so that the whole time of $n$ vibrations is

where $T$ is the time deduced from the observations. Hence, to find the time $T$ in infinitely small arcs, we have approximately,

To find the time $T_{0}$ when there is no damping, we have

$T_{0}$	${}=T_{1}\sin \alpha$
	${}=T_{1}{\frac {\pi }{\sqrt {\pi ^{2}+\lambda ^{2}}}}$.

741.] The equation of the rectilinear motion of a body, attracted to a fixed point and resisted by a force varying as the velocity, is

where $x$ is the coordinate of the body at the time $t$, and $a$ is the coordinate of the point of equilibrium.