Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/374

 

To solve this equation, let

the solution of which is

		$y=C\cos \left({\sqrt {\omega ^{2}-k^{2}}}t+a\right)$, when $k$ is less than $\omega$;		(4)
		$y=A+Bt$, when $k$ is equal to $\omega$;		(5)
and		$y=C^{\prime }\cos h\left({\sqrt {k^{2}-\omega ^{2}}}t+a^{\prime }\right)$, when $k$ is greater than $\omega$.		(6)

The value of $x$ may be obtained from that of $y$ by equation (2). When $k$ is less than $\omega$, the motion consists of an infinite series of oscillations, of constant periodic time, but of continually decreasing amplitude. As $k$ increases, the periodic time becomes longer, and the diminution of amplitude becomes more rapid.

When $k$ (half the coefficient of resistance) becomes equal to or greater than $\omega$, (the square root of the acceleration at unit distance from the point of equilibrium,) the motion ceases to be oscillatory, and during the whole motion the body can only once pass through the point of equilibrium, after which it reaches a position of greatest elongation, and then returns towards the point of equilibrium, continually approaching, but never reaching it.

Galvanometers in which the resistance is so great that the motion is of this kind are called dead beat galvanometers. They are useful in many experiments, but especially in telegraphic signalling, in which the existence of free vibrations would quite disguise the movements which are meant to be observed.

Whatever be the values of $k$ and $\omega$, the value of $a$, the scale-reading at the point of equilibrium, may be deduced from five scale-readings, $p$, $q$, $r$, $s$, $t$, taken at equal intervals of time, by the formula

$a={\frac {q(rs-qt)+r(pt-r^{2})+s(qr-ps)}{(p-2q+r)(r-2s+t)-(q-2r+s)^{2}}}$.

On the Observation of the Galvanometer.

742.] To measure a constant current with the tangent galvanometer, the instrument is adjusted with the plane of its coils parallel to the magnetic meridian, and the zero reading is taken. The current is then made to pass through the coils, and the deflexion of the magnet corresponding to its new position of equilibrium is observed. Let this be denoted by $\phi$.

Then, if $H$ is the horizontal magnetic force, $G$ the coefficient of the galvanometer, and $\gamma$ the strength of the current,