left to oscillate. The period was observed in some cases by noting the times of the successive transits of the needle across the vertical cross wire of an observation telescope; but the method finally adopted was to attach to the stirrup as shown in Fig. 4 a light silvered mirror m (*3 cm. in diameter and or gramme in mass), and to use the same lamp and scale as in the deflection experiments. This latter arrangement enabled the amplitude of oscillation to be reduced to less than a degree and so reduced to zero the correction necessary for arc. The moment of inertia of the mirror was only about 40000 of that of the deflector, and its neglect therefore introduced an error of only 400 per cent.

Time was observed in these experiments by means of a very accurate watch provided with a centre seconds hand moving round a dial divided into quarter seconds. When two observers were available, one counted the oscillations and called sharply "Now" at the end of every four or five periods, while the other observed the time at each call. When only one observer counted the oscillations he used a chronometer beating half seconds. Having read time, he counted the beats until he could observe a transit. He then counted the beats until he observed another transit. From the result he estimated the number of periods in one minute, and therefore observed the time of the first transit after each minute so long as there was sufficient amplitude. The fractions of half seconds were estimated from the positions of the magnet at the beat next before and the beat next after the transit. With the mirror and scale arrangement these observations could be made with great accuracy.

The observations were combined in the following manner so as to give the most probable value of the

...

period. Supposing the number of observations to have been even, 2n say. The interval between the nth observation and the (n + 1)th, three times that between the (n - 1)th and the (+2)th, five times that between the (n − 2)th and the (n + 3)th, and so on to that between the 1st and the 27th were added together, the sum divided by the sum of the series 12+ 32 + 52 + + (2n − 1)2, and the result by the number of periods (which was the same in each case) between each successive pair of observations. This gave the average period to a high degree of approximation. If an odd number of observations (2n + 1) was taken, the interval between the nth and the (2 + 2)th, twice that between the (n − 1)th and the (n + 3)th, three times that between the (1⁄2 — 2)th and the (n + 4)th, and so on to the 1st and (2n + 1)th, were added together, the sum divided by twice the sum of the series 12+22+ 32 + . . . + n2, and the result divided by the number of periods in each interval gave the average period. The period adopted was always the mean of those given by two closely agreeing sets of observations.

Assuming (see p. 36) that the magnet has two definite poles, that is (in this connection) points at which the whole of the free magnetism in each half of the magnet may be supposed concentrated in considering the external action of the magnet (an assumption not seriously erroneous in the case of the thin magnets and the distances used); the distance between them can be calculated from the results of deflection experiments in the side-on and end-on positions obtained as described above, since the effect of the distribution is opposite in the two cases. For if r be the distance for the end-on position, the deflection, and r', 'the distance and deflection for the side-on

position, we have by equating the values of M/H given by equations (1) and (2):

Expanding the numerator and denominator of each side and neglecting terms smaller than those of the second order we get:

230 22'30'

2r0 + 3r'0'

(9)

By this equation the value of λ used in the calculation of H and M was found. The results for magnets of different lengths and diameters are interesting in themselves.

The moment of inertia of the bar was found by weighing the bar and carefully measuring its length and crosssection, and calculating for a vertical axis through the centre of the magnet supposed hung horizontally. The axis of suspension of the magnet in any case was not, however, that vertical, but another near it owing to the compensation for the tendency of the magnet to dip in the earth's field. The distance between these two axes can be found approximately for each magnet from the magnetic moment, mass, and length as given in the table below, and is so small that any error caused by supposing the magnet simply to vibrate round the former vertical is well within the possible limit of accuracy.

For a cylindrical magnet of mass W, actual length 2/ and diameter d, the moment of inertia is W (12/3 +ď2/64). Hence (3) becomes :

Hence for a single deflector we get instead of equations

(4), (5), (6), (7) equations obtained from these by substituting instead of l2, 12 + 3ď2/64.

If two deflectors be used, each of the actual length 27, and diameter d, but of masses W1, W2, periods T1, T2, and nearly equal effective lengths which give a mean λ, we get from (1) and (2) instead of (5) and (6) for the end-on and side-on positions respectively:

In these formulas

and are the angular deflections found from the mean readings taken as described above (p. 24).

There are two corrections for alteration of moment of the magnet, produced (1) by variation of temperature, (2) by induction when the magnet is in or near the magnetic meridian when oscillating. The first correction was found by placing the magnet within a bath, in one of two principal positions at such a distance from the magnetometer needle that a deflection of 1,000 divisions was obtained, and then raising the temperature through about 40° C. It was found that such a rise of temperature produced a change of deflection of only about two divisions. Thus the magnets changed in magnetic moment by only 200 per cent. for a change of temperature of 1° C. Hence as the variation of temperature in the experiments never exceeded 2° C. or 3° C. this correction was neglected.

The correction for induction was found by immersing the deflecting magnet in an artificially produced magnetic field of known strength, and ascertaining the alteration of

magnetic moment which resulted. The field was produced by surrounding the magnet with a magnetizing coil, and its intensity calculated from the number of turns of wire per unit of length of the coil and the current-strength, which was measured. The coil was sufficiently long to project beyond the magnet at each end some distance, so that the magnetic field was uniform, and equal to 4C, where n number of turns per cm. of length, and C the current strength in C.G.S. units (see below, Chap. III.).

=

Fig. 5 shows the arrangement of apparatus for these experiments; m is the magnetometer needle, C, C' are coils each consisting of silk-covered copper wire wound on glass tubes 5 cm. in external diameter, S is the lamp scale, R a box of resistance coils, G the current galvanometer, Ka reversing key, and B a battery. DE represents a horizontal line through the needle and in the magnetic meridian, and AF a horizontal line at right angles to DE, and also passing through the centre of the needle. As