Page:Scientific Memoirs, Vol. 2 (1841).djvu/87

tion of the earth's magnetism. But this very circumstance affords the means of an easy solution. Let us suppose that in our figure the straight line, from the centre of the magnet $N\;S$, through the needle $n\;s$, coincides with the magnetic meridian; in this position the terrestrial magnetic force will not act at all on the needle $n\;s$. As soon, however, as the moment of rotation which $N\;S$ exerts on $n\;s$ begins to act, $n\;s$ will be deflected from its original position, and set in motion; but the more it deviates on account of this movement from its first position, the more strongly does the earth's magnetism tend to bring it back to its former position. The needle consequently performs vibrations about a line, which is no longer in the direction of the magnetic meridian itself, but is more or less inclined to it. This line is the position of equilibrium of the needle $n\;s$, which it assumes when the vibrations have ceased. This direction is evidently that of the resultant of the two forces, viz. the earth's magnetism, and the magnetism of the needle $N\;S$. According to the well-known laws of statics, the proportion of the strength of these forces, which is also the proportion of the moments of rotation produced by them, may consequently be determined from the angle of deviation, i. e. from the difference between the two positions of repose of $n\;s$, when it is subjected to the action of both the forces; and when $N\;S$ is removed.

"Here then arises another important remark; namely, that the angle of deviation of the needle $n\;s$ is quite independent of its magnetism; as any increase in that respect evidently causes both moments of rotation to increase in the same proportion. We are thus freed from the necessity of fulfilling the difficult condition of equality in the magnetism of the two needles."

 

If we represent the deflection by $v$,—the greatest moment of rotation exerted by the earth on the needle (according to the measure fixed for the terrestrial magnetism) by $m\;T$,—and by $F$, the moment of rotation exerted by the magnetism of the bar ($M$) on the magnetism of the needle ($m$) at the distance $R$; the forces exerted by the earth, and by the bar, on the needle, are to each other in the proportion of the cosine to the sine of the deflection $v$; and the moments of rotation, $mT$ and $F$ being also in the same relation to each other,

$mT:F::\cos v:\sin v,$

i.e.

$mT={\frac {F}{\mathrm {tang} \,v}}.$

(III.)