4. The method which mathematicians adopt in treating of simultaneous motions is to consider them one after the other. A velocity is the describing of a certain length or angle in a certain time. Properties of small linear and angular displacements are properties of linear and angular velocities. If, therefore, a body is subject to two independent motions, as rotations about two axes or a rotation and a translation, it is considered to obey them in turn each for a very short time. A rifle bullet moving towards the target and rotating all the time is supposed to approach the target without turning, through an infinitely small space, and then to turn round through an infinitely small angle, much like a man descending a spiral staircase. This is not the actual motion, any more than a polygon is a curve, but it differs as little as we please from the real motion, and it clears our ideas and enables us to apply mathematical methods to the problems.

If a body is solicited to two different motions by two simultaneous causes, it will in reality follow neither; but it may be supposed to have followed both.

Thus the very extravagant idea of some of the earliest writers on projectiles that a cannon-ball went straight until it had exhausted the force of projection and then fell down straight under gravity, had in it, notwithstanding its grievous confusion of force with velocity, a germ of truth, (viz.) that the causes of motion must be considered separately.

A skater describing circles, the nut of a screw, a crank rod one end of which moves in a straight line while the other describes a circle, the arms of a common form of reaping machine which rotate about an inclined axis while carried forwards by the machine, and hundreds of other familiar cases, supply examples of translation combined with

rotation.

Examples of combined rotations are seen in the screw or paddles of a steamer, which are rotating about a horizontal axis while the steamer may be moving round a curve

and thus rotating about the vertical; or in the common gyroscopic toy, where a metal ring rotates about a diameter of a circle, and is borne along also by the rotation of this circle about the vertical; or in the sails of a windmill, which may have rotations about their own axis, about the vertical (if the wind veers) and with the whole body of the windmill round the polar axis of the earth.

5. Motions in one plane and in parallel planes.

Let A be a point of a rigid body which is moved to A'. Bisect AA' perpendicularly by the straight line NX. On

NX take any point B. We may suppose that the point A has been moved to A' by the body having been caused to rotate about an axis through B at right angles to the plane of the paper. In this case the line of particles AB has taken up the position A'B. Now cause the body to rotate about an axis through A' perpendicular to the plane of the paper through an equal and opposite angle. The line A'B takes up the position A'B' parallel to AB, whence we infer that a displacement of translation is equivalent to two equal and opposite displacements of rotation about two parallel axes. If these displacements are small, AA' is at right angles to AB, and the proposition becomes that two equal and opposite angular velocities about two parallel axes are equivalent to a translational velocity in a direction at right angles to the plane of the axes.

It will be convenient to denote such axes perpendicular to the plane of the paper by the point where they cut this plane. Thus we might have spoken of A' and B as axes.

This important proposition may be reduced to the parallelogram of linear velocities.

Let P be any point of a body which has simultaneous equal and opposite rotations round A and B. The velocities of P

a

b

A

due to these are represented by Pa and Pb perpendicular and proportional to PA and PB respectively. The resultant velocity is represented by the diagonal PR of the parallelogram on Pa, Pb. But this parallelogram is similar to that whose sides are PA, PB. Hence the velocity of Pis proportional and perpendicular to AB. As this holds for every point of the body, the whole is being translated at right angles to AB, and with velocity proportional to it.

6. From the definition of rotation it is clear that two equal and opposite rotations cannot produce a rotation; for they turn a straight line in the body through equal but opposite angles.

For the same reason the angular velocity of a body rotating about two parallel axes is the sum or difference of their separate angular velocities,

Let P be any point of a body which has angular velocities in the same direction round A and B. And take on PB a point C such that PB and PC are proportional to the angular velocities round A and B respectively. Then the linear

velocities of P will be at right angles to PA and PB, and proportional to PA. PB and PB. PC respectively, i. e. to PA

R

and PC. Hence, joining AC and bisecting it in N, the resultant velocity of Pis at right angles to PN and is measured by twice PN. P may therefore be taken to be revolving about any point in PN. Let PN produced cut AB in R; we can shew that R is the same point wherever P is, and therefore the whole body is rotating about R. The angular velocity has been settled independently.

Produce CP to C', making PC' equal to PC. Join AC'. PNR is parallel to AC, and therefore

AR: RB: C'P: PB;

or R divides AB inversely as the angular velocities round A and B.

7. Any displacement whatever may be given to a body by a translation and a rotation about an axis.

For to bring AB to A'B' it is only necessary to translate. the body till a point A reaches its new position A', and then to rotate the body about A'.

Thus in general any motion of a body may be composed of a rotation round an arbitrary point and a translation.

And this point may generally be so chosen that the movement of translation shall not be required. In other words, there is one point which is unaffected by the change of position. To find this point. Bisect AA' and BB' perpendicularly, and let the bisectors meet in N.

Join NA, NA', NB, NB'.

N

B'

A

Then NA is equal to NA' and NB to NB'.

If, then, we can shew that the angle ANA' is equal to the angle BNB', we shall have proved that when the body is rotated about N, so as to move A to A', B is brought to B', and so for any other point. For N is a point of the body, since the triangles ANB and A'NB' are equal in all respects.

Now the triangles ANB, A'NB', have all their sides equal each to each; therefore the angles ANB and A'NB' are equal. Take away the common angle A'NB, and the remainder ANA' is equal to the remainder BNB'.

Hence any motion of a rigid body, except one of translation, is one of rotation round some axis; and this is called the instantaneous axis.

If the body be a plane one, and be moving in its plane, this axis cuts the plane in the instantaneous centre. To find its position it is only necessary to take two points whose directions of motion are known, and to draw perpendiculars from them to these directions. Their intersection gives the centre required.