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(1) the only effect it can produce is a horizontal motion about the vertical axis.

4. When one end of the axis of the revolving disc is supported by a string, and the other end left without a support, the disc does not fall, but revolves horizontally round the string as a centre. The action of gravity tends to turn the disc round a horizontal axis, and we have seen (1) that the rotating disc resists all forces which tend to alter its plane of rotation. When the speed is sufficiently great it resists the force of gravity, therefore it cannot fall; but it moves horizontally round the string, because the action of gravity tends to turn the disc round a horizontal axis, as in Exp. 3, and its motion in a horizontal direction round the string is owing to the same cause.

5. When the disc is inclined to one side, a motion frequently ensues round a vertical axis. This is caused by the friction of the pivots on the outer ring. When the axis of the disc is perfectly horizontal, the friction has no tendency to produce this motion. When the motion commences, the inclination of the disc continually increases, because the rotation of the disc tends to carry any point of the circumference in the direction of a tangent, and the motion round a vertical axis turns the plane of the disc out of that direction, the inertia of the point assumed acts as described in (2). This increases the inclination, the friction now acts at a greater advantage, thus the motion about the vertical axis is accelerated, which again increases the inclination of the disc, bringing it still nearer the horizontal position. At length this position is attained, and now the whole force of friction is employed in producing a motion round the vertical axis.

6. When the gyroscope is fastened by a screw or clamp in such a manner as to keep the rotating disc always vertical, and placed in the centre of a turning table, it maintains its plane of rotation unchanged, notwithstanding

the motion of the table. When placed near the edge of the table, so as to be conveyed through the circumference of a circle, the plane of the disc is always parallel to its first position, for the reason stated in (1). If it were placed in the centre of a circle at the pole of the earth, the plane of the disc would appear to move round a vertical axis in twenty-four hours. This property of maintaining the plane of rotation unchanged has been proposed as a means of proving experimentally the rotation of the earth on its axis. If the instrument be placed at any point between the pole and the equator, the time of an apparent revolution round a vertical axis will be more than twenty-four hours, and the time is greater the nearer it approaches to the equator. When at the equator it will not move round the vertical axis at all.

In order to explain this satisfactorily, it is necessary to state what is meant by saying that the revolving disc maintains its plane of rotation unchanged. The plane of rotation is not absolutely unchangeable, since it always coincides with a great circle of the earth (when clamped as described) passing through the station, and therefore it partakes of the diurnal motion of the earth, except when situated at one of the poles.

Let A (Fig.11) be the station, and let it be carried through the small arc AB by the diurnal motion of the earth, the plane of rotation of the disc at the point B, is not BY, which is a small circle, but BDV, a great circle intersecting the great circle APV in two points VV', diametrically apposite.

Since APV = 90°, and AP is the distance of A from the pole P, therefore PV = latitude of A. And when the aro AB is very small, BDV = APV = 90°. Hence, in the spherical triangle VPB, (Fig. 4.)

Sin VPB : sin PBV = sin VB : sin VP. or Sino : sin e

= R: sin lat. or Arc ab : arc c d = 1 : sin lat.

Now, ab is the arc of a circle which the plane of rotation of the disc would appear to describe if placed at the pole, and cd is the arc which it would appear to describe in the same time in the latitude AB; and since the arcs are to each other inversely as the times of description of the whole circumference, it follows that the time of describing the whole circumference in latitude A : time of describing the whole circumference at P = ab: cd. =1: sin lat. A.

Time at P 24 hours
Time in lat A =

sin lat. sin lat.

24 hours and Time at Equator

= Infinity

O As I am by no means certain of the correctness of some of the conclusions at which I have arrived, I submit them to the Society with considerable diffidence and distrust, as subjects for discussion rather than as demonstrated truths. They seem to me, at present, to follow legitimately from the doctrine of compound rotatory motion. If there is any error in my reasoning, I am not able to discover it; and I shall feel greatly obliged to any one who will indicate where it is, as I shall thus be enabled to approach nearer the truth than the point at which I have arrived by my own independent investigations.

Professor Elliot thanked Professor Hamilton for the more than ample justice he had dealt out to him in noticing his instruments. He begged leave to make some additional statements regarding his own researches on the subject, as well as those of others. Professor Hamilton had correctly described his (Professor Elliot's) instruments for illustrating, both in regard to fact and principle, the precession of the equinoxes, the nutation of FIG. II


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