Let a body be rotating simultaneously about OA and OB with velocities proportional to OA and OB.

N

.B

Then, first, the points on OC are at rest. For, taking such a point P and drawing PM, PN at right angles to OA and OB, P's linear velocity due to its rotation round OA is upwards from the plane of the paper, and proportional to OA.PM; while that due to rotation round OB is downwards and proportional to OB. PN. As these products are twice the areas of the triangles OPA, OPB respectively, they are equal. As they are opposite the point P is at rest. The body is therefore rotating about OC.

To settle its angular velocity about OC; draw a perpen

dicular to the plane AOB through O, and on this take a point Q.

The angular velocities of the body round OA, OB will be proportional to the linear velocities of perpendicular to these, i.e. Oa, Ob, which are drawn in the plane of AOB, at right angles to OA, OB and proportional to them respectively.

The resultant angular velocity about OC will be represented by the resultant linear velocity of Q, i.e. by Oc, the diagonal of the parallelogram Oa, Ob. But the parallelograms OA, OB and Оa, Ob are similar, the latter being turned round through a right angle. Therefore the diagonal Oc is proportional and perpendicular to OC. And the resultant angular velocity of the body is proportional to OC.

2. Angular velocities, then, are quantities which obey the parallelogram law, and all its consequences will hold good for them. A body rotating with velocity w about any axis may be considered to have a component angular velocity a cosa about any other axis inclined to the former at an angle a. There will be a parallelopiped of angular velocities; and in general the analogy between angular velocities and forces in Statics is complete.

We will take for the illustration of this the pendulum experiment by Foucault, by which the rotation of the earth is rendered visible.

Draw a circle representing a section of the earth through its polar axis NS. Let O be the centre, and A any place on

its surface.

In this experiment, a pendulum is set swinging in any vertical plane at A. We assume that wherever the point of suspension may be, the plane in which the pendulum swings will remain parallel to itself. If the earth were rotating about OA, the effect of this would be that the plane of the pendulum would be left behind by the earth, and would appear to an observer, unconscious of the earth's motion, to follow the sun. Now this is in part what happens. The earth does not indeed rotate about OA; its rotation is about NS; but this is equivalent to one about OA proportional to cos NOA, and one about a perpendicular to OA

proportional to sin NOA. This latter is what carries the building and the whole apparatus eastward. It does not affect the present question. But the other rotation-that about 04-causes the plane of the pendulum to follow that of the sun with an angular velocity, which is to that of the earth as cos NOA to 1.

This experiment requires the greatest care for its exhibition. If the pendulum move in even the most elongated oval, instead of swinging in a plane, the axis of this oval will rotate from a very different cause, viz. the resistance of the air. When Foucault exhibited his experiment in Paris to the French 'savants,' he used a heavy ball hung from the roof of the Observatory, and set it off by burning a thread which held the ball out of the position of rest.

3. If a rigid body has one point fixed, there is at any moment a straight line of points at rest. In other words, any displacement of a rigid body, one point of which is fixed, may be effected by a single rotation about some axis through that point. The proof is the same as that by which we shewed that any displacement of a plane body in its plane can be given by a rotation round one point; if instead of a plane we consider a sphere in the body with the fixed point as centre. The points, then, represent straight lines through the centre; the straight lines in the figure become arcs of great circles and represent planes passing through the centre; but the reasoning is precisely the same.

Any displacement may therefore be given to a rigid body by translating it so that a chosen point comes into its new position, and then making it rotate round some axis through that point. The direction of this axis and the angular displacement remain the same whatever point be chosen. The direction and amount of translation may change, but the translation cannot affect the angular movement.

The point may be chosen so that the direction of translation is that round which the rotation takes place. For let C' be the new position of C, and let C'X be the axis of rotation. Let AB represent a plane in the body perpendicular to this axis. Let A'B' be its final position. This is parallel

to AB; for neither the translation nor the rotation about C'X affects its direction.

If, then, we first translate the whole body along a parallel to CX until AB comes to A'B' in the same plane with "B", we shall be able by one rotation about a parallel to C'X to bring A'B' to A"B", i.e. the direction of translation will be the axis of rotation.

Hence every small motion is reducible to that of a screw in its nut. And all points of any rigid body are at the same moment moving in coaxial helices. If the pitch of the screw be zero the motion will be one of rotation simply, if it be infinite it will be a translation.

It is of course not always equally easy to see what these axes and directions are. In the case of a rifle bullet, for example, the motion is already reduced. In the general case the first point is to find out the series of planes which remain parallel, or-what is the same thing-to find the direction of rotation. Thus suppose we are considering the motion of the earth at any instant. This consists of a rotation round its polar axis and a revolution of its centre in the plane of the ecliptic round the sun. And suppose we wish to reduce it to the screw motion. We observe that the planes which remain parallel are those parallel to the equator. Hence the axis of the screw is perpendicular to the equator. To find the actual position of this axis we must consider all velocities projected on a plane parallel to the equator. Then the motion is similar to that of the hoop in Lesson 1. Art. 8, which rotates about its centre while the centre revolves about a fixed point.

Let be the velocity of rotation of the earth, V the component velocity of its centre in the plane of the equator, R the distance between two axes each perpendicular to the equator, through the centres of the sun and earth respectively. By Lesson I. Art. 8, V and Q are equivalent to two angular velocities, about the axis through the sun, and

Ω

V
R

V

R

about the axis through the centre of the earth. And these are equivalent to an angular velocity round a parallel V axis in the plane of the others, distant R- from that Ω through the sun. This last is therefore the axis of rotation.

4. An extremely elegant geometrical conception of the motion of a body round a fixed point was introduced by Poinsot. Any such motion may be completely represented by imagining a cone fixed in the moving body to roll on a cone fixed in space. For every body with one point fixed is rotating about a certain axis. As the motion changes, this axis takes up different positions, and describes a cone whose vertex is the fixed point. Now by reasoning exactly similar to that of Lesson I. Art. 8, any cone rolling on another with the same vertex has for its instantaneous axis its line of contact with the other. This axis therefore describes a cone whose vertex is the fixed point. But this is precisely the motion to be represented. The rate of rotation will depend on the dimensions of the rolling cone.

As an example of this take the case of a top spinning with angular velocity w about its axis OC, while that axis is rotating with angular velocity Q about the vertical OV, to which it is inclined at an angle a. By the parallelogram of angular velocities the resultant axis is OR-between OC and OV—inclined to OC at an angle given by

Hence the motion is the same as if an imaginary right circular cone in the top, whose axis was the axis of the top, and whose semivertical angle was ROC, were to roll on the

P. G.

2