Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/372

 

$t_{1}$ is the observed time of the first transit of the mark in the positive direction, and $t_{2}$, $t_{3}$, &c. the times of the following transits.

If $T$ be the time of vibration, and $P_{1}$, $P_{2}$, $P_{3}$, &c. the times of transit of the true point of equilibrium,

$t_{1}=P_{1}+{\frac {x}{v_{1}}}$, $t_{2}=P_{2}+{\frac {x}{v_{2}}}$,

where $v_{1}$, $v_{2}$, &c. are the successive velocities of transit, which we may suppose uniform for the very small distance $x$. If $\rho$ is the ratio of the amplitude of a vibration to the next in succession,

$v_{2}=-{\frac {1}{\rho }}v_{1}$, and${\frac {x}{v_{2}}}=-\rho {\frac {x}{v_{1}}}$.

If three transits are observed at times $t_{1}$, $t_{2}$, $t_{3}$, we find

${\frac {x}{v_{1}}}={\frac {t_{1}-2t_{2}+t_{3}}{(\rho +1)^{2}}}$.

The period of vibration is therefore

$T={\frac {1}{2}}(t_{3}-t_{1})-{\frac {1}{2}}{\frac {\rho -1}{\rho +1}}(t_{1}-2t_{2}+t_{3})$.

The time of the second passage of the true point of equilibrium is

$P_{2}={\frac {1}{4}}(t_{1}+2t_{2}+t_{3})-{\frac {1}{4}}{\frac {(\rho -1)^{2}}{(\rho +1)^{2}}}(t_{1}-2t_{2}+t_{3})$.

Three transits are sufficient to determine these three quantities, but any greater number may be combined by the method of least squares. Thus, for five transits,

$T={\frac {1}{10}}(2t_{5}+t_{4}-t_{2}-2t_{1})-{\frac {1}{10}}(t_{1}-2t_{2}+2t_{3}-2t_{4}+t_{5}){\frac {\rho -1}{\rho +1}}\left(2-{\frac {\rho }{1+\rho ^{2}}}\right)$.

The time of the third transit is,

$P_{3}={\frac {1}{8}}(t_{1}+2t_{2}+2t_{3}+2t_{4}+t_{5})-{\frac {1}{8}}(t_{1}-2t_{2}+2t_{3}-2t_{4}+t_{5}){\frac {(\rho -1)^{2}}{(\rho +1)^{2}}}$.

739.] The same method may be extended to a series of any number of vibrations. If the vibrations are so rapid that the time of every transit cannot be recorded, we may record the time of every third or every fifth transit, taking care that the directions of successive transits are opposite. If the vibrations continue regular for a long time, we need not observe during the whole time. We may begin by observing a sufficient number of transits to determine approximately the period of vibration, $T$, and the time of the middle transit, $P$, noting whether this transit is in the positive or the negative direction. We may then either go on counting the vibrations without recording the times of transit, or we may leave the apparatus unwatched. We then observe a