jewel and pivot, and, in addition, costs far less; but with a fibre suspension it is generally necessary that the instrument should have levelling screws, such as are seen attached to G, Fig. 56, and that it should be levelled before being used.

A galvanometer needle should therefore be supported by a pivot when the instrument has to be moved about, and used quickly in different positions. But when the galvanometer is employed in

a fixed position, and great accuracy is desired, the needle ought always to be suspended by a fibre of unspun silk. For the exact method of preparing a silk fibre and using it to suspend a needle see the Appendix, page 566.

Fig. 57.

31. Sine Law. Another interesting case of the combination of the controlling and deflecting forces (Fig. 41, § 21, page 81) occurs when the controlling force, N P, is constant in magnitude, but not necessarily in direction; and when the direction of the deflecting force, PR, instead of making a constant angle with the controlling force N P, as in the tangent galvanometer, makes a constant angle with the direction of the deflected body.

As already has been explained, equilibrium of a needle, whose two ends are symmetrical with the coil, and which is controlled by a uniform magnetic field, will occur when the four points R, N, S, R' (Fig. 42, page 81) are in one straight line. Hence, the condition that the deflecting force PR (Fig. 41, page 81) makes a constant angle with the needle, is equivalent to saying that the angle P R N is a constant.

Now, if a perpendicular, P U, be let fall from P on to NR (Fig. 57), the sine of PN R, that is the sine of the

angle the needle makes with the direction of the controlling force, equals the ratio of PU to NP, while the sine of P R N equals the ratio of P U to P R. Hence

but by hypothesis, P N is constant in magnitude, so also is sin. PR N, therefore

PR is proportional to sin. P N R.

But since P R, the deflecting force, makes a constant angle with the needle, the needle must make a constant angle with the coil. And this is the condition we saw in § 19, page 70, which causes the deflecting force to be proportional to the current. Hence we know that the deflecting force is proportional to sin. P N R, also that the deflecting force is proportional to the current, therefore the current must be proportional to sin. P N R, that is to the sine of the angle the needle makes with the direction of the controlling force.

Generally, then, we may say that if a body, turning on an axis, be acted on by two forces in the plane in which the body is free to turn, the deflecting force will be proportional to the sine of the angle between the body and the controlling force if:

(1st) The controlling force is constant in magnitude, but not necessarily in direction.

(2nd) The angle between the direction of the deflecting force and the deflected body is kept constant.

If then there be a rod N N' (Fig. 58)-short or longturning about a pivot o, and acted on by a weight w (and we apply various deflecting forces by placing weights w' in the scale pan), and, if further, after the application of each weight, w', we alter something so as to bring the angle between the rod and the direction of the deflecting force-that is, between N N' and NQalways to the same value, these deflecting forces will be proportional to the sines of the angles that N N' makes with N P, the direction of the controlling force w.

There are three distinct ways in which, after the application of different deflecting forces, w', the angle between N N', and N Q, the direction of the deflecting force, can be brought always to the same value.

(1st) By altering the direction of the deflecting force without altering its magnitude.

(2nd) By altering the direction of the controlling force without altering

its magnitude.

(3rd) By altering

the magnitude of the controlling force.

con

The last method is inadmissible, as stancy in the magnitude of the controlling force is a condition for the sine law being true. But either methods (1) or (2), or a combination of them, may be employed.

The apparatus seen in Fig. 59 is arranged for utilising the first of the methods, by causing the angle between N N' and the direction of the

Fig. 58.

deflecting force, to have the same value in any one set of experiments. On inserting different weights w' in the pan, the pivoted rod N N' will be pulled more or less to one side, and the angle between it and the deflecting force will be altered. But, if for each weight the screw T be turned, and the arms o D and EQ be revolved together round a centre o, until the pivoted rod N N' has the same position relatively to o D, it will have the same position relatively to N Q, the direction of the deflecting force.

When making a set of measurements, the first thing to do is to remove the silk thread, kk, off the pulley p, and place the scale pan on the base board of the apparatus; then rest the weight w on a block of wood, or hold it in the hand, so that both the deflecting and

the controlling forces are removed. The counterpoise weight c is then screwed in or out until the rod N N' remains balanced in any position.

Next the controlling weight w is allowed to pull the rod N N' vertical, and the scale ss is adjusted, so that the silk thread supporting w, and the reflection of this thread in the piece of looking-glass attached to the scale, are seen coinciding with the zero on the scale.

A weight w' having been placed in the scale pan, the screw T is turned until the reflection of a projecting point at the lower end of N N' (seen in a small piece of lookingglass G carried at the end of the arm o D) coincides with a scratch on this glass. This device enables N N', after the insertion of each weight w' in the scale pan, to be very accurately caused to have the same position rela tively to o D, and

therefore relatively

to N Q, the direction of the deflecting force.

The angle which

N N' makes with the direction of the controlling force is NOU (Fig. 60), and its sine the ratio of U N

to N O. But the length NO remains constant as N N' turns about o, therefore the sine is proportional to UN or Z P, that is to the

distance the silk thread supporting the controlling weight w has moved along the scale s s from its zero z.

By loosening the nut n (Fig. 59), turning the arm E Q relatively to the arm o D, and tightening the nut n again, EQ can be fixed so as to make any angle with o D. Further, by sliding the screw and nut n along the slot, which is on the end of the arm o D, the end of the arm EQ can be fixed at various distances from o; and experiment shows that in whatever initial position the arm EQ may be fixed relatively to the arm o D, the distances ZP corresponding with a set of deflecting weights w' are proportional to these weights, provided that after the insertion of each weight in the pan the screw T be