Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/343

706.] From the general equation of $$M$$, Art. 703, we obtain another set of conditions, &c.;

Solving these equations and substituting the values of the coefficients, the series for $$M$$ becomes

706.] Omitting the corrections of Art. 705, we find by Art. 673 where $$n$$ is the number of windings of the wire, $$a$$ is the mean radius of the coil, and $$R$$ is the geometrical mean distance of the transverse section of the coil from itself. See Art. 690. If this section is always similar to itself, $$R$$ is proportional to its linear dimensions, and $$n$$ varies as $$R^2$$.

Since the total length of the wire is $$2 \pi an$$, $$a$$ varies inversely as $$n$$. Hence and we find the condition that $$L$$ may be a maximum