Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/370

 

part of the motion of another point which moves with uniform angular velocity in a logarithmic spiral.

733.] The equation of the spiral is

$r=Ce^{-\phi \cot \alpha }$,

To determine the horizontal motion, we put

$\phi =\omega t$,$x=a+r\sin \phi$,

where $a$ is the value of $x$ for the point of equilibrium.

If we draw $BSD$ making an angle $\alpha$ with the vertical, then the tangents $BX$, $DY$, $GZ$, &c. will be vertical, and $X$, $Y$, $Z$, &c. will be the extremities of successive oscillations.

734.] The observations which are made on vibrating bodies are—

(1) The scale-reading at the stationary points. These are called Elongations.

(2) The time of passing a definite division of the scale in the positive or negative direction.

(3) The scale-reading at certain definite times. Observations of this kind are not often made except in the case of vibrations of long period^[1].

The quantities which we have to determine are—

(1) The scale-reading at the position of equilibrium.

(2) The logarithmic decrement of the vibrations.

(3) The time of vibration.

To determine the Reading at the Position of Equilibrium from Three Consecutive Elongations.

735.] Let $x_{1}$, $x_{2}$, $x_{3}$ be the observed scale-readings, corresponding to the elongations $X$, $Y$, $Z$, and let $a$ be the reading at the position of equilibrium, $S$, and let $r_{1}$ be the value of $SB$,

$x_{1}-a={}$	$r_{1}\sin \alpha$,
$x_{2}-a=-{}$	$r_{1}\sin \alpha e^{-\pi \cot \alpha }$,
$x_{3}-a={}$	$r_{1}\sin \alpha e^{-2\pi \cot \alpha }$.

From these values we find

$(x_{1}-a)(x_{3}-a)=(x_{2}-a)^{2}$,

whence $a={\frac {x_{1}x_{3}-x_{2}^{2}}{x_{1}+x_{3}-2x_{2}}}$.

When $x_{3}$ does not differ much from $x_{1}$ we may use as an approximate formula

$a={\frac {1}{4}}(x_{1}+2x_{2}+x_{3})$.

1. See Gauss, Resultate des Magnetischen Vereins, 1836. II.