Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/351

715.] The field of force due to the double coil is represented in section in Fig. XIX at the end of this volume.

714.] By combining four coils we may get rid of the coefficients $$G_2$$, $$G_3$$, $$G_4$$, $$G_5$$, and $$G_6$$. For by any symmetrical combinations we get rid of the coefficients of even orders Let the four coils be parallel circles belonging to the same sphere, corresponding to angles $$\theta$$, $$\phi$$, $$\pi - \phi$$, and $$\pi - \theta$$.

Let the number of windings on the first and fourth coil be $$n$$, and the number on the second and third $$pn$$. Then the condition that $$G_3 = -$$ for the combination gives Rh and the condition that $$G_5 =0$$ givesRh

Puttingand expressing $$Q_3^\prime$$ and $$Q_5^\prime$$ (Art. 698) in terms of these quantities, the equations (1) and (2) become RhRh

Taking twice (4) from (5), and dividing by 3, we get Rh

Hence, from (4) and (6), and we obtain

Both $$x$$ and $$y$$ are the squares of the sines of angles and must therefore lie between 0 and 1. Hence, either $$x$$ is between 0 and $$\frac{4}{7}$$, in which case $$y$$ is between $$\frac{6}{7}$$ and 1, and $$p$$ between $$\infty$$ and $$\frac{49}{32}$$, or else $$x$$ is between $$\frac{6}{7}$$ and 1, in which case $$y$$ is between 0 and $$\frac{4}{7}$$, and $$p$$ between 0 and $$\frac{32}{49}$$.

715.] The most convenient arrangement is that in which $$x = 1$$. Two of the coils then coincide and form a great circle of the sphere whose radius is $$C$$. The number of windings in this compound coil is 64. The other two coils form small circles of the sphere. The radius of each of them is $$\sqrt{\frac{4}{7}}C$$. The distance of either of