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

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715.]
GALVANOMETER OF THREE COILS.
319

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 , , , , and . 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 , , , and .

Let the number of windings on the first and fourth coil be , and the number on the second and third . Then the condition that for the combination gives

,
(1)

and the condition that gives

,
(2)
Putting
and ,
(3)

and expressing and (Art. 698) in terms of these quantities, the equations (1) and (2) become

(4)
.
(5)

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

.
(6)

Hence, from (4) and (6),

,

and we obtain

, .

Both and are the squares of the sines of angles and must therefore lie between 0 and 1. Hence, either is between 0 and , in which case is between and 1, and between and , or else is between and 1, in which case is between 0 and , and between 0 and .


Galvanometer of Three Coils.

715.] The most convenient arrangement is that in which . Two of the coils then coincide and form a great circle of the sphere whose radius is . 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 . The distance of either of