The angle d may now be found by means of a pro

tractor.

Example 18.--If the horizontal component of the earth's magnetism in 1893 at London be the controlling force in a tangent galvanometer, the bobbin of which is 11 inches in diameter, how many convolutions of wire must be wound on the bobbin in order that a current of 1.017 ampere may give a deflection of 45° ?

Answer. 4 convolutions.

Example 19.--If the horizontal component of the earth's magnetism in 1895 at London be the controlling force in a tangent galvanometer, the bobbin of which is wound with eight convolutions of wire, what must be the radius of the bobbin in order that a current of 0.384 ampere may give a deflection of 50° ?

Answer.-3.48 inches. Tan. 50° may be found either in a table of tangents or in the following way ::

Take a sheet of squared paper; on it take axes o x, o y, at right angles to one another; with a protractor make the angle BOX, equal to 50°, and produce o B as far as the paper will allow. Let A B be the farthest line from o, parallel to oy, which cuts B 0. Then

Count the number of divisions and fractions of division in a B and o A, and divide the one by the other. If the angle be large, great care must be taken to lay it down accurately with the protractor, since a small error in a large angle will introduce a large error in the tangent.

Example 20.-About how many times the horizontal component of the earth's magnetism must the controlling force be in a tangent galvanometer, having a bobbin 5 inches in radius wound with six convolutions of wire, in order that a current of 20 amperes may make a deflection of 45° ?

Answer. About 32 times.

Example 21.-The needle of a tangent galvanometer when acted on by the earth's field alone makes one oscillation in 13 second, whereas, when the controlling magnet is placed in position, it makes one oscillation in 0.433 second. If the coil be half a foot in radius, and be wound with twenty turns of wire, what current will produce a deflection of 30° in 1898 at London?

Answer.-1.16 ampere.

It is not necessary that the coil of a tangent galvanometer should be circular, but in order to obtain the straightness of the lines of force in the neighbourhood of the axis, as seen in Figs. 34, 38, and 53, and not merely for points actually on the axis, of which we could only avail ourselves by using an infinitely short magnet, the diameter of all parts of the coil must be large. Hence, if an elliptic or other non-circular coil were used, its smallest diameter would have to be large, and consequently its largest diameter unnecessarily so.

28. Magnetometer. The following is an important example of the use of the tangent law. Let ns (Fig. 55) be a small compass needle, suspended so as to be free to turn in a horizontal plane, and first let it be acted on only by a uniform magnetic field, produced by the

earth for example. Along a horizontal line passing through the centre of the needle, and perpendicular to the position taken up by the needle when acted on by the uniform field alone, let a magnet N s be placed, the distance of the nearest end N of this magnet from the needle being considerable compared with the length of the needle itself.

We have then the exact conditions for the tangent law to be true, and, consequently, the tangent of the

angle through which the needle n s turns in taking up the deflected position n's' is a measure of the magnetic force exerted by the large magnet N S at the spot occupied by the little magnet.

Such an arrangement constitutes what is called a "magnetometer," and it may be used to test the strength of different magnets of the same length, put successively so as to occupy the position NS; or, by making a single magnet, NS, occupy different positions along the line AB, and measuring the tangent of the different deflections of the needle n s, we can find out the force produced by a given magnet at different positions along its axial line A B.

29. Calibrating any Galvanometer by Direct Comparison with a Tangent Galvanometer. The necessity that a galvanometer in order to obey the tangent law should have its coil very large, compared with the length of the needle, prevents a tangent galvanometer from being very sensitive. It also renders a tangent galvanometer unportable, for if the needle were made very short instead of the coil being made large, its movement would be seriously impeded by the mass of even a very light pointer attached to it. If the indications of an instru ment are to be unaffected by moving it from one place to another, as well as unaffected by the proximity of a mass of iron like a fireplace, iron water-pipes, &c., the controlling magnet must be powerful, must be attached to the galvanometer, and have its poles close to the needle. Hence, a uniform controlling field which, as has been seen, is one of the conditions for the tangent law to be true, cannot be attained in such an instrument.

The law of such a galvanometer must, therefore, be obtained experimentally, and a very convenient way of performing the calibration is to compare the deflections of the instrument under test with those of a tangent galvanometer, when the same currents pass through both apparatus, which may be arranged as in Fig. 56.

G is the standard tangent galvanometer, D the

galvanometer, which, if rough and portable, is sometimes called a 66 detector," requiring to be calibrated. v is a Vshaped tube containing two zinc rods dipping into a small quantity of a solution of zinc sulphate, and is used for varying the strength of the currents passing through G and D by altering the distance between the bottoms of the zinc rods. The wires coming from the currentgenerator are attached to the terminals TT, and a key

G

placed between G and D enables the current to be made or broken. As the same current passes through G and D, it is quite unnecessary to know the value of the resistance introduced by v; all that has to be done is to observe a number of corresponding deflections of the needles of G and of D, then, since the true value of the current is proportional to the tangent of the deflection in G, a calibration curve can be drawn for D, in which horizontal distances represent the observed angular deflection of the needle of D, and vertical distances the relative strengths of the currents producing these deflections. If the number of amperes producing any particular

deflection in G be also known, then D will be calibrated absolutely.

It frequently happens that, on account of the great increase in sensitiveness produced by putting the wires conveying the current close to the needle, a rough galvanometer with a few turns of wire is more sensitive than a tangent galvanometer with many turns. Under such circumstances it would be difficult to compare them, as a large deflection on D would only correspond with a small one on G, and a smaller deflection on D would not produce a deflection on a large enough to be read at all accurately. This difficulty may, however, be overcome by putting a piece of wire s (Fig. 56), to act as a bye path, a "shunt" as it is called, between the terminals of D, which shunt allows a portion of the current to pass through it instead of through D. As, however, for the same shunt the same fraction of the total current is, as we shall see later on ($ 96, page 289), always shunted past D, the sensibility alone of D, and not the law connecting current strength with deflection, is altered by using such a shunt. The use of a shunt, therefore, alters the absolute but not the relative calibration of a galvanometer; consequently, if D is absolutely calibrated, the same shunt must always be employed when it is desired to use the absolute calibration curve of that galvanometer.

30. Pivot and Fibre Suspensions.-The galvanometers G and D differ also in another particular, namely, in the way in which the magnetic needle is supported. In D the little magnet has a jewel in its centre, and rests on a sharp pivot, as in an ordinary pocket compass; whereas in G the needle is supported by a fine fibre of unspun silk, the upper end of which is fastened in one of the ways illustrated in Fig. 25 (page 49), so that it can be lowered on to the card s s, on which the scale is engraved, when the instrument is being carried about, and raised again so as to be in the centre of the coil when the instrument is in use. The fibre suspension introduces far less friction to the motion of the needle than the best