whirl round on my toes like a ballet-dancer while this is in my hand. I move it about in all sorts of ways, but if it was pointing to the pole star at the beginning it remains pointing to the pole star; if it pointed towards the moon at the beginning it still points

towards the moon. The fact is, that as there is almost no frictional constraint at the pivots there are almost no forces tending to turn the axis of rotation of the gyrostat, and I can only give it motions of translation. But now I will clamp this vertical spindle by means of a screw and repeat my ballet-dance whirl; you will note that I need not whirl round, a very small portion of a whirl is enough to cause this gyrostat (Fig. 18) to set its spinning axis vertical, to set its axis parallel to the vertical axis of rotation which I give it. Now I whirl in the opposite direction, the gyrostat at once turns a somersault, turns completely round and remains again with its axis vertical, and if you were to carefully note the direction of the spinning of the

gyrostat, you would find the following rule to be generally true:—Pay no attention to mere translational motion, think only of rotation about axes, and just remember that when you constrain the axis of a spinning body to rotate, it will endeavour to set its own axis parallel to the new axis about which you rotate it; and not only is this the case, but it will endeavour to have the direction of its own spin the same as the direction of the new rotation. I again twirl on my toes, holding this frame, and now I know that to a person looking down upon the gyrostat and me from the ceiling, as I revolved in the direction of the hands of a clock, the gyrostat is spinning in the direction of the hands of a clock; but if I revolve against the clock direction (Fig. 19) the gyrostat tumbles over so as again to be revolving in the same direction as that in which I revolve.

Fig. 19.

This then is the simple rule which will enable you to tell beforehand how a gyrostat will move

when you try to turn it in any particular direction. You have only to remember that if you continued your effort long enough, the spinning axis would become parallel to your new axis of motion, and the direction of spinning would be the same as the direction of your new turning motion.

Now let me apply my rule to this balanced gyrostat. I shove it, or give it an impulse downwards, but observe that this really means a rotation about the horizontal axis C D (Fig. 13), and hence the gyrostat turns its axis as if it wanted to become parallel to C D. Thus, looking down from above (as shown by Fig. 20), O E was the direction of the spinning axis, O D was the axis about which I endeavoured to move it, and the instantaneous effect was that O E altered to the position O G. A greater impulse of the same kind would have caused the spinning axis instantly to go to O H or O J, whereas an upward opposite impulse would have instantly made the spinning axis point in the direction O K, O L or O M, depending on how great the impulse was and the rate of spinning. When one observes these phenomena for the first time, one says, "I shoved it down, and it moved to the right; I shoved it up, and it moved to the left;" but if the direction of the spin were opposite to what it is, one would say, "I shoved it down, and it moved to the left; I shoved it up, and it moved to the right." The simple

statement in all cases ought to be, "I wanted to rotate it about a new axis, and the effect was to send its spinning axis towards the direction of the new axis." And now if you play with this balanced gyrostat as I am doing, shoving it about in all sorts of ways, you will find the rule to be a correct one, and there is no difficulty in predicting what will happen.

Fig. 20.

If this rule is right, we see at once why precession takes place. I put this gyrostat (Fig. 13) out of balance, and if it were not rotating it would fall downwards; but a force acting downwards really causes the gyrostat to move to the right, and so you see that it is continually moving in this way, for the force is always acting downwards, and the spinning axis is continually chasing the new axes about which gravity tends continually to make it revolve. We see also why it is that if the want of balance is the other way, if gravity tends to lift the gyrostat, the precession is in the opposite direction. And in playing with this gyrostat as I do now, giving it all sorts of pushes, one makes other observations and sees that the above rule simplifies them all; that is, it enables us to remember them. For example, if I use this stick to hurry on the precession, the gyrostat moves in opposition to the force which causes the precession. I am particularly anxious that you should remember this. At present the balance-weight is so placed that the gyrostat would fall if it were not spinning. But it is spinning, and so it precesses. If gravity were greater it would precess faster, and it comes home to us that it is this precession which enables the force of gravity to be inoperative in mere downward motion. You see that if the precession is hurried, it is more than sufficient to balance gravity,

and the gyrostat rises. If I retard the precession, it is unable to balance gravity, and the gyrostat falls. If I clamp this vertical axis so that precession is impossible, you will notice that the gyrostat falls just as if it were not spinning. If I clamp the instrument so that it cannot move vertically, you notice how readily I can make it move horizontally; I can set it rotating horizontally like any ordinary body.

In applying our rule to this top, observe that the axis of spinning is the axis E F of the top (Fig. 12). As seen in the figure, gravity is tending to make the top rotate about the axis F D, and the spinning axis in its chase of the axis F D describes a cone in space as it precesses. This gyrostat, which is top-heavy, rotates and precesses in much the same way as the top; that is, if you apply our rule, or use your observation, you will find that to an observer above the table the spinning and precession occur in the same direction, that is, either both with the hands of a watch, or both against the hands of a watch. Whereas, a top like this before you (Fig. 21), supported above its centre of gravity, or the gyrostat here (Fig. 22), which is also supported above its centre of gravity, or the gyrostat shown in Fig. 56, or any other gyrostat supported in such a way that it would be in stable equilibrium if it were not spinning; in all these

cases, to an observer placed above the table, the precession is in a direction opposite to that of the spinning.