Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/339

703.]

An expression for $$M$$, which is sometimes more convenient, is got by making $$c_1 = \frac{r_1 - r_2}{r_1 + r_2}$$ in which case Rh

702.] The lines of magnetic force are evidently in planes passing through the axis of the circle, and in each of these lines the value of $$M$$ is constant.

Calculate the value of $$K_\theta = \frac{\sin \theta}{(F_{\sin \theta} - E_{\sin \theta})^2}$$ from Legendre's tables for a sufficient number of values of $$\theta$$.

Draw rectangular axes of $$x$$ and $$z$$ on the paper, and, with centre at the point $$x = \frac{1}{2} a (\sin \theta + \mathrm{cosec}\, \theta)$$, draw a circle with radius $$\frac{1}{2} a (\mathrm{cosec}\, \theta - \sin \theta)$$. For all points of this circle the value of $$c_1$$ will be $$\sin \theta$$. Hence, for all points of this circle, Rh

Now $$A$$ is the value of $$x$$ for which the value of $$M$$ was found. Hence, if we draw a line for which $$x = A$$, it will cut the circle in two points having the given value of $$M$$.

Giving $$M$$ a series of values in arithmetical progression, the values of $$A$$ will be as a series of squares. Drawing therefore a series of lines parallel to $$z$$, for which $$x$$ has the values found for $$A$$, the points where these lines cut the circle will be the points where the corresponding lines of force cut the circle.

If we put $$m = 4 \pi a$$, and $$M = nm$$, then Rh

We may call $$n$$ the index of the line of force.

The forms of these lines are given in Fig. XVIII at the end of this volume. They are copied from a drawing given by Sir W. Thomson in his paper on 'Vortex Motion '.

703.] If the position of a circle having a given axis is regarded as defined by $$b$$, the distance of its centre from a fixed point on the axis, and $$a$$, the radius of the circle, then $$M$$, the coefficient of induction of the circle with respect to any system whatever