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

 

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

Galvanometer of Four Coils.

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

and the condition that $G_{5}=0$ gives

and expressing $Q_{3}^{\prime }$ and $Q_{5}^{\prime }$ (Art. 698) in terms of these quantities, the equations (1) and (2) become

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

Hence, from (4) and (6),

$p={\frac {x}{y}}{\frac {5x-4}{4-5y}}={\frac {x^{2}}{y^{2}}}{\frac {7x-6}{6-7y}}$,

and we obtain

$y={\frac {4}{7}}{\frac {7x-6}{5x-4}}$, $p={\frac {32}{49x}}{\frac {7x-6}{(5x-4)^{3}}}$.

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 ${\frac {32}{49}}$.

Galvanometer of Three Coils.

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