It is my hope that the book may be useful not only to students of Natural Philosophy, but also to engineers. Most of them possess a knowledge of the principles of Mechanics, of the method of Co-ordinate Geometry, and of the Integral Calculus; and that is all that is here required.

The principle on which this science is based has been so long connected with the name of D'Alembert that it would hardly be recognised under any other. Nevertheless there is no doubt that Euler has more claim to its authorship, inasmuch as he first used it. D'Alembert admits this, but says that Euler gave no proof. I believe D'Alembert's real merit to be, that his explanation was exactly suited to clear away the difficulties which were perplexing men's minds.

The works to which I am principally indebted are:Thomson and Tait's Natural Philosophy; Routh's Rigid Dynamics; Resal's Cinématique Pure; Rankine's Machinery and Millwork; Walton's Mechanical Problems; Whewell's History of the Inductive Sciences; Willis' Principles of Mechanism; Müller's Lehrbuch der kosmischen Physik; Montucla's Histoire des Mathématiques; D'Alembert's Traité de Dynamique, and Euler's Mechanik.

My thanks are due to Dr Campion, of Queens' College, for many valuable suggestions which he has made; and to several of my pupils for their frank statement of their difficulties.

G. PIRIE.

QUEENS' COLLEGE, CAMBRIDGE.
December, 1874.

GEOMETRY OF MOTION.

I.

1. A RIGID body is an assemblage of particles such that the distance between each pair is unchangeable. The movements of such a body are very different from those of a set of independent points. Its fixed connec tions introduce a common movement. Any straight line or any plane of particles in the body must remain always a straight line or plane. If all such planes remain parallel to themselves, the motion is one of translation. But if any such plane makes an angle with its former position the motion is rotational. And the velocity of rotation-angular velocity is measured by the rate at which the plane is describing angles.

Thus the connecting rod of the driving wheels of a goods' locomotive has only a translational motion;-so also (approximately) the axis of the earth in its yearly motion round the sun. In a well-thrown quoit the motions are combined.

2. From this definition of rotation it follows that a point cannot rotate. It may revolve about another point, but it contains no lines nor planes which can describe angles. For rotation there must be an extended system. A point in motion may be said to be revolving about any point whatever situated in the line through it at right angles to its direction of motion, for it is moving at the moment in a circle with the point as centre. But the body of which this is a point may not be rotating. For rotation it is necessary that the different points of the body should be at the moment revolving about the same axis,

P. G.

1

Suppose a man to move round a column viewing its parts in succession. In this case he is also rotating. Were he to move without rotation, he must work round the column sometimes forwards, sometimes sidewards, sometimes backwards, but always facing the same point of the compass.

3. From the definition it follows also that rotation is directional, i. e. it takes place not so much about an axis or point

A

B'

B

as about a direction or in a plane. The body AB has rotated in passing to the position A'B'; but the amount of the rotation is measured by the angle between the straight lines AB and A'B'. It matters not to the angular velocity of a carriage-wheel whether it is rolling along a level road or up a hill, or whether the wheel, being raised from the ground, is whirled round its own axle. Or on a larger scale, whether a ship rounds a promontory or swings with her anchor fixed through the same angle is indifferent to the amount and direction of the rotation performed. There is indeed in general an axis round which the body may, in a stricter sense, be said to rotate, for every point of the body moves in a circle about it. This rotation is the more easily imagined, but it ought not to be allowed to expel the idea of the other. It is a pity there are not separate words to distinguish them. We will in future speak of them as rotation round a point or axis, and rotation round a direction or in a plane.