MAGNETISM 239 time T of its vibration under the earth s force, we obtain the product KH, p, say. Secondly, by the method of deflexion, of which two varieties, tangent deflexion and sine deflexion, are in use, the value of the quotient K/H, =q, say, is found. In this method K is used as the deflecting magnet, and the moment K of the deflected magnet does not appear in the result. 1 It is obvious that, if we know the value of H, or may assume it constant, either of these methods will enable us to express the moment of any magnet in terms of that of another arbitrarily chosen as unit; and, reciprocally, if we operate with a magnet of known or of constant moment, we can determine the values of H at different times and places in terms of its value at an arbitrarily chosen time and place. By combining two observations, in one of which a magnet K is the vibrating and in the other the deflecting magnet, we can obtain both K and H in absolute measure, for we have two equations KH = p> and K/H = q, which give K = /pq , and H = "vp/q . (Jnifilar Vibration Experiments. If 6 be the angle between the axis of magneto- the magnet and H at time t, y the angle between the axis and meter: H in the position of no torsion, rKH the coefficient of torsion, vibration then the equation of motion of the magnet, when the arc of oscilla- experi- tion is very small, may be written A0+KHfl + TKH(0-7)--0 (59). This gives for the period of a complete vibration T 2 = 47T 2 A/KH(l+r) (60). The observations are made with the magnetometer arranged as for the declination experiment. The swinging magnet is brought to rest, and the circle so clamped that the axial point of the magnet scale is on the cross wire of the telescope; the magnet is then slightly disturbed so as to oscillate through a small arc (16 or so). The time of vibration is found first roughly, by taking the time of a single vibration, then more accurately by counting a large number of vibrations and timing the end of the last as accurately as possible. T is found by observing the de flexion 9 and 6" caused by turning the torsion head through an angle /3 in one direction and then through an angle /3 in the opposite direction; we thus get from equation (59) KHe + T KH(0 - 7 - j8) = , ments. There are several corrections which, although in general negligible, may sometimes require to be considered. (1) H may vary so much during the experiment as to cause a sensible error; (2) if the arc of vibration be too large, it may be necessary to apply the reduction to infinitely small arcs; (3) if the amplitude of the vibrations decrease too rapidly, account must be taken of the resistance to the motion arising from the viscosity of the air, &c.; 2 (4) a correction has to be made for the alteration of the moment of the magnet by the earth s induction, 3 and (5) a temperature correction for the magnetic moment and the moment of inertia. Deflexion Experiments. Gauss s arrangement the deflecting Deflexion magnet was placed in an east- west direction, i.e., end on to the experi- original position of the deflected magnet. The equation of ments equilibrium in this case is [see equation (51)] T T. TT/ - . f. n t 2K.lv 1 1 P KH(l + T)sm0=cos0 -. 7 + i + I I ^J ^.4 J.5 I with unifdar magneto meter. (61). where P J = T 1 /2K K, P 2 =T. 2 /2K K, &c. In the method of sines the deflecting magnet is turned until it is perpendicular to the axis of the deflected magnet in its final position of equilibrium; the equation of equilibrium in this case is r 3 H 2K (62). The advantage of the method of tangents is that the moment of the deflector is not affected inductively by the earth s force. In the method of sines a "correction has on this account to be made; but, on the other hand, there is no torsion, and, from the symmetry of the position of the two magnets, the approximate formula) have a more exact application. The new pattern of the unifilar magnetometer is adapted for the method of sines. The instrument arranged as in fig. 29 is first carefully levelled, and fitted with the graduated cross bar D, which and r = (0 - From the same equation we may also determine y when necessary. The most troublesome part of the whole process still re mains, viz., the determination of A. This is effected by attaching to the magnet a body whose moment of inertia B can be calculated from its dimensions. For this purpose Gauss fixed a cross bar of wood to the magnet, and attached to it at known equal distances from the axis of suspension two cylindrical weights of known mass and dimension. Sometimes a cylinder of gun metal is slung below the magnet by means of two loops. Perhaps the best method is to use a ring of gun metal attached to the magnet so that its plane is horizontal and its centre as nearly as possible in the line of suspen sion. The new time of vibration being T : , and the new coefficient of torsion (if different) T , we have the new equation From this and (60) we get A = B 1+ 1 This important fact was first noticed by Lambert. FIG. 29. Unifilar Magnetometer, arranged to show deflexion. " is so set in its sockets as to be perpendicular to the line of collima- tion of the telescope A. The box is opened, and the torsion removed from the suspending fibre by means of a plummet as already explained. The deflected magnet is then suspended so as to be at the same height as the deflecting magnet wh, n the latter is placed in its carriage on the cross bar. The sides of the box are now closed, and the circle of the instrument turned until the middle division of the scale B, seen by reflexion from a mirror attached to the deflected magnet, is on the cross wires of the telescope A; tl circle is then clnmped, and the verniers read. The deflecting map K (the same as that used in the vibration experiments) is next placed in its carriage L on the cross bar at a distance r, (30 cm. or so) east; the circle is then turned until the middle division of the scale is again on the cross wires; the verniers are read once more, difference between the two readings being lt we have _ 2 In this connexion see more especially Lament. Handb. d. Mag- netismtis, pp. 282 sq. .
3 See Lamont, Handb., p. 371; Maxwell, vol. in. 457.