Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/133

455.] If we suppose that one of these distances, say the smaller, is given, the value of the greater distance may be determined so as to make $$\delta Q$$ a minimum. This condition leads to an equation of the fifth degree in $$r_1^2$$, which has only one real root greater than $$r_2^2$$. From this the best value of $$r_1$$ is found to be $$r_1=1.3189r_2$$.

If one observation only is taken the best distance is when

where $$\delta D$$ is the probable error of a measurement of deflexion, and $$\delta r$$ is the probable error of a measurement of distance.

Method of Sines.

455.] The method which we have just described may be called the Method of Tangents, because the tangent of the deflexion is a measure of the magnetic force.

If the line $$r_1$$ instead of being measured east or west, is adjusted till it is at right angles with the axis of the deflected magnet, then $$R$$ is the same as before, but in order that the suspended magnet may remain perpendicular to $$r$$, the resolved part of the force $$H$$ in the direction of $$r$$ must be equal and opposite to $$R$$. Hence, if $$\theta$$ is the deflexion, $$R=H\sin\theta$$.

This method is called the Method of Sines. It can be applied only when $$R$$ is less than $$H$$.

In the Kew portable apparatus this method is employed. The suspended magnet hangs from a part of the apparatus which revolves along with the telescope and the arm for the deflecting magnet, and the rotation of the whole is measured on the azimuth circle.

The apparatus is first adjusted so that the axis of the telescope coincides with the mean position of the line of collimation of the magnet in its undisturbed state. If the magnet is vibrating, the true azimuth of magnetic north is found by observing the extremities of the oscillation of the transparent scale and making the proper correction of the reading of the azimuth circle.

The deflecting magnet is then placed upon a straight rod which passes through the axis of the revolving apparatus at right angles to the axis of the telescope, and is adjusted so that the axis of the deflecting magnet is in a line passing through the centre of the suspended magnet.

The whole of the revolving apparatus is then moved till the line