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

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306
CIRCULAR CURRENTS.
[701.

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 for two parallel circles by elliptic integrals.

For let 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 and be the radii of the circles, then

,

the integration being extended round both curves. In this case,

,

, , ,

,
where
,

and and are complete elliptic integrals to modulus . From this we get, by differentiating with respect to and remembering that is a function of ,

.

If and denote the greatest and least values of ,

, ,

and if an angle be taken such that ,

,

where and denote the complete elliptic integrals of the first and second kind whose modulus is . If , , and

.

The quantity represents the attraction between two parallel circular currents, the current in each being unity.