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We have had considerable difficulty in compiling this treatise from the larger work—arising from the necessity for condensation to a degree almost incompatible with the design to omit nothing of importance: and we feel that it would have given us much less trouble and anxiety, and would probably have ensured a better result, had we written the volume anew without keeping the larger book constantly before us. The sole justification of the course we have pursued is that wherever, in the present volume, the student may feel further information to be desirable, he will have no difficulty in finding it in the corresponding pages of the larger work.

A great portion of the present volume has been in type since the autumn of 1863, and has been printed for the use of our classes each autumn since that date.

W. THOMSON,
P. G. TAIT..

November, 1872.

DIVISION I.

PRELIMINARY.

CHAPTER 1.-KINEMATICS.

1. The science which investigates the action of Force is called, by the most logical writers, DYNAMICS. It is commonly, but erroneously, called MECHANICS ; a term employed by Newton in its true sense, the Science of Machines, and the art of making them. . 2. Force is recognized as acting in two ways:

1° so as to compel rest or to prevent change of motion, and

2° so as to produce or to change motion. Dynamics, therefore, is divided into two parts, which are conveniently called STATICS and KINETICS.

3. In Statics the action of force in maintaining rest, or preventing change of motion, the balancing of forces,' or Equilibrium, is investigated; in Kinetics, the action of force in producing or in changing motion.

4. In Kinetics it is not mere motion which is investigated, but the relation of forces to motion. The circumstances of mere motion, considered without reference to the bodies moved, or to the forces producing the motion, or to the forces called into action by the motion, constitute the subject of a branch of Pure Mathematics, which is called KINEMATICS, or, in its more practical branches, MECHANISM.

5. Observation and experiment have afforded us the means of translating, as it were, from Kinematics into Dynamics, and vice versa. This is merely mentioned now in order to show the necessity for, and the value of, the preliminary matter we are about to introduce.

6. Thus it appears that there are many properties of motion, displacement, and deformation, which may be considered altogether independently of force, mass, chemical constitution, elasticity, temperature, magnetism, electricity; and that the preliminary consideration of such properties in the abstract is of very great use for Natural Philosophy. We devote to it, accordingly, the whole of this chapter;

WISY.

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which will form, as it were, the Geometry of the subject, embracing what can be observed or concluded with regard to actual motions, as long as the cause is not sought. In this category we shall first take up the free motion of a point, then the motion of a point attached to an inextensible cord, then the motions and displacements of rigid systems—and finally, the deformations of solid and fluid masses.

7. When a point moves from one position to another it must evidently describe a continuous line, which may be curved or straight, or even made up of portions of curved and straight lines meeting each other at any angles. If the motion be that of a material particle, however, there can be no abrupt change of velocity, nor of direction unless where the velocity is zero, since (as we shall afterwards see) such would imply the action of an infinite force. It is useful to consider at the outset various theorems connected with the geometrical notion of the path described by a moving point; and these we shall now take up, deferring the consideration of Velocity to a future section, as being more closely connected with physical ideas.

8. The direction of motion of a moving point is at each instant the tangent drawn to its path, if the path be à curve; or the path itself if a straight line. This is evident from the definition of the tangent to a curve.

9. If the path be not straight the direction of motion changes from point to point, and the rate of this change, per unit of length of the curve, is called the Curvature. To exemplify this, suppose

two 'tangents PT, QU, drawn to a circle, and radii OP, OQ, to the points of contact. The angle between the tangents is the change of direction between P and l, and the rate of change is to be measured by the relation between this angle and the length of the circular arc PQ. Now, if o

be the angle, s the arc, and r the radius, we see at once that (as the angle between the radii is equal to the angle between the tangents, and as the measure of an angle is the ratio of the arc to the radius, $ 54)

r0 = 8, and therefore -=- is the measure of the curvature. Hence the cur

S vature of a circle is inversely as its radius, and is measured, in terms of the proper unit of curvature, simply by the reciprocal of the radius.

10. Any small portion of a curve may be approximately taken as a circular arc, the approximation being closer and closer to the truth, as the assumed arc is smaller. The curvature at any point is the reciprocal of the radius of this circle for a small arc on each side of the point.

11. If all the points of the curve lie in one plane, it is called a plane

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