Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/131

454·] Taking the arithmetical mean of (6) and (7),

Now remove $$M$$ to the west side of the suspended magnet, and place it with its centre at the point marked $$2s_0-s$$ on the scale. Let the deflexion when the axis is in the first position be $$\theta_3$$, and when it is in the second $$\theta_4$$, then, as before,

Let us suppose that the true position of the centre of the suspended magnet is not $$s$$ but $$s_0+\sigma$$, then

and

and since $${\theta_2}{r_2}$$ may be neglected if the measurements are carefully made, we are sure that we may take the arithmetical mean of $${r_1}^n$$ and $${r_2}^n$$ for $$r^n$$.

Hence, taking the arithmetical mean of (8) and (9),

or, making

454.] We may now regard $$D$$ and $$r$$ as capable of exact determination.

The quantity $$A_2$$ can in no case exceed $$2L^2$$, where $$L$$ is half the length of the magnet, so that when $$r$$ is considerable compared with $$L$$ we may neglect the term in $$A_2$$ and determine the ratio of $$H$$ to $$M$$ at once. We cannot, however, assume that $$A_2$$ is equal to $$2L^2$$, for it may be less, and may even be negative for a magnet whose largest dimensions are transverse to the axis. The term in $$A_4$$, and all higher terms, may safely be neglected.

To eliminate $$A_2$$, repeat the experiment, using distances $$r_1,r_2,r_3,$$ &c., and let the values of $$D$$ be $$D_1, D_2, D_3$$, &c., then