Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/338

306 is moderate as compared with the radii of the smaller, the series already given do not converge rapidly. In every case, however, we may find the value of $$M$$ for two parallel circles by elliptic integrals.

For let $$b$$ be the length of the line joining the centres of the circles, and let this line be perpendicular to the planes of the two circles, and let $$A$$ and $$a$$ be the radii of the circles, thenthe integration being extended round both curves.

In this case, whereand $$F$$ and $$E$$ are complete elliptic integrals to modulus $$c$$.

From this we get, by differentiating with respect to $$b$$ and remembering that $$c$$ is a function of $$b$$, Rh

If $$r_1$$ and $$r_2$$ denote the greatest and least values of $$r$$, and if an angle $$\gamma$$ be taken such that $$\cos \gamma = \frac{r_2}{r_1}$$, Rhwhere $$F_\gamma$$ and $$E_\gamma$$ denote the complete elliptic integrals of the first and second kind whose modulus is $$\sin \gamma$$.

If $$A = a$$, $$\cot \gamma = \frac{b}{2a}$$, and

The quantity $$\frac{dM}{db}$$ represents the attraction between two parallel circular currents, the current in each being unity.