forces. Thus the velocities acquired in one second by the same mass (falling freely) at different parts of the earth's surface, give us the relative amounts of the earth's attraction at these places.

Again, if equal forces be exerted on different bodies, the changes of velocity produced in equal times must be inversely as the masses of the various bodies. This is approximately the case, for instance, with trains of various lengths started by the same locomotive: it is exactly realized in such cases as the action of an electrified body on a number of solid or hollow spheres of the same external diameter, and of different metals.

Again, if we find a case in which different bodies, each acted on by a force, acquire in the same time the same changes of velocity, the forces must be proportional to the masses of the bodies. This, when the resistance of the air is removed, is the case of falling bodies; and from it we conclude that the weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass; a most important physical truth, which will be treated of more carefully in the chapter devoted to Properties of Matter.

225. It appears, lastly, from this law, that every theorem of Kinematics connected with acceleration has its counterpart in Kinetics. Thus, for instance (§ 38), we see that the force under which a particle describes any curve, may be resolved into two components, one in the tangent to the curve, the other towards the centre of curvature; their magnitudes being the acceleration of momentum, and the product of the momentum and the angular velocity about the centre of curvature, respectively. In the case of uniform motion, the first of these vanishes, or the whole force is perpendicular to the direction of motion. When there is no force perpendicular to the direction of motion, there is no curvature, or the path is a straight line.

226. We have, by means of the first two laws, arrived at a definition and a measure of force; and have also found how to compound, and therefore also how to resolve, forces: and also how to investigate the motion of a single particle subjected to given forces. But more is required before we can completely understand the more complex cases of motion, especially those in which we have mutual actions between or amongst two or more bodies; such as, for instance, attractions, or pressures, or transferrence of energy in any form. This is perfectly supplied by

227. LEX III. Actioni contrariam semper et aequalem esse reactionem : sive corporum duorum actiones in se mutuò semper esse aequales et in partes contrarias dirigi.

To every action there is always an equal and contrary reaction: or, the mutual actions of any two bodies are always equal and oppositely directed. 228. If one body presses or draws another, it is pressed or drawn by this other with an equal force in the opposite direction. If any one presses a stone with his finger, his finger is pressed with the same force in the opposite direction by the stone. A horse towing a boat on a canal is dragged backwards by a force equal to

that which he impresses on the towing-rope forwards. By whatever amount, and in whatever direction, one body has its motion changed by impact upon another, this other body has its motion changed by the same amount in the opposite direction; for at each instant during the impact the force between them was equal and opposite on the When neither of the two bodies has any rotation, whether before or after impact, the changes of velocity which they experience are inversely as their masses.

two.

When one body attracts another from a distance, this other attracts it with an equal and opposite force. This law holds not only for the attraction of gravitation, but also, as Newton himself remarked and verified by experiment, for magnetic attractions: also for electric forces, as tested by Otto-Guericke.

229. What precedes is founded upon Newton's own comments on the third law, and the actions and reactions contemplated are simple forces. In the scholium appended, he makes the following remarkable statement, introducing another specification of actions and reactions subject to his third law, the full meaning of which seems to have escaped the notice of commentators :

Si aestimetur agentis actio ex ejus vi et velocitate conjunctim; et similiter resistentis reactio aestimetur conjunctim ex ejus partium singularum velocitatibus et viribus resistendi ab earum attritione, cohaesione, pondere, et acceleratione oriundis; erunt actio et reactio, in omni instrumentorum usu, sibi invicem semper aequales.

In a previous discussion Newton has shown what is to be understood by the velocity of a force or resistance; i. e. that it is the velocity of the point of application of the force resolved in the direction of the force, in fact proportional to the virtual velocity. Bearing this in mind, we may read the above statement as follows :-

If the action of an agent be measured by the product of its force into its velocity; and if, similarly, the reaction of the resistance be measured by the velocities of its several parts into their several forces, whether these arise from friction, cohesion, weight, or acceleration;-action and reaction, in all combinations of machines, will be equal and opposite.

Farther on we shall give a full development of the consequences of this most important remark.

230. Newton, in the passage just quoted, points out that forces of resistance against acceleration are to be reckoned as reactions equal and opposite to the actions by which the acceleration is produced. Thus, if we consider any one material point of a system, its reaction against acceleration must be equal and opposite to the resultant of the forces which that point experiences, whether by the actions of other parts of the system upon it, or by the influence of matter not belonging to the system. In other words, it must be in equilibrium with these forces. Hence Newton's view amounts to this, that all the forces of the system, with the reactions against acceleration of the material points composing it, form groups of equilibrating systems for these points considered individually. Hence, by the

principle of superposition of forces in equilibrium, all the forces acting on points of the system form, with the reactions against acceleration, an equilibrating set of forces on the whole system. This is the celebrated principle first explicitly stated, and very usefully applied, by D'Alembert in 1742, and still known by his name. We have seen, however, that it is very distinctly implied in Newton's own interpretation of his third law of motion. As it is usual to investigate the general equations or conditions of equilibrium, in treatises on Analytical Dynamics, before entering in detail on the kinetic branch of the subject, this principle is found practically most useful in showing how we may write down at once the equations of motion for any system for which the equations of equilibrium have been investigated.

231. Every rigid body may be imagined to be divided into indefinitely small parts. Now, in whatever form we may eventually find a physical explanation of the origin of the forces which act between these parts, it is certain that each such small part may be considered to be held in its position relatively to the others by mutual forces in lines joining them.

232. From this we have, as immediate consequences of the second and third laws, and of the preceding theorems relating to centre of inertia and moment of momentum, a number of important propositions such as the following:

(a) The centre of inertia of a rigid body moving in any manner, but free from external forces, moves uniformly in a straight line.

(6) When any forces whatever act on the body, the motion of the centre of inertia is the same as it would have been had these forces been applied with their proper magnitudes and directions at that point itself.

(c) Since the moment of a force acting on a particle is the same as the moment of momentum it produces in unit of time, the changes of moment of momentum in any two parts of a rigid body due to their mutual action are equal and opposite. Hence the moment of momentum of a rigid body, about any axis which is fixed in direction, and passes through a point which is either fixed in space or moves uniformly in a straight line, is unaltered by the mutual actions of the parts of the body.

(d) The rate of increase of moment of momentum, when the body is acted on by external forces, is the sum of the moments of these forces about the axis.

233. We shall for the present take for granted, that the mutual action between two rigid bodies may in every case be imagined as composed of pairs of equal and opposite forces in straight lines. From this it follows that the sum of the quantities of motion, parallel to any fixed direction, of two rigid bodies influencing one another in any possible way, remains unchanged by their mutual action; also that the sum of the moments of momentum of all the particles of the two bodies, round any line in a fixed direction in space, and

passing through any point moving uniformly in a straight line in any direction, remains constant. From the first of these propositions we infer that the centre of inertia of any number of mutually influencing bodies, if in motion, continues moving uniformly in a straight line, unless in so far as the direction or velocity of its motion is changed by forces acting mutually between them and some other matter not belonging to them; also that the centre of inertia of any body or system of bodies moves just as all their matter, if concentrated in a point, would move under the influence of forces, equal and parallel to the forces really acting on its different parts. From the second we infer that the axis of resultant rotation through the centre of inertia of any system of bodies, or through any point either at rest or moving uniformly in a straight line, remains unchanged in direction, and the sum of moments of momenta round it remains constant if the system experiences no force from without. This principle is sometimes called Conservation of Areas, a not very convenient designation.

234. The kinetic energy of any system is equal to the sum of the kinetic energies of a mass equal to the sum of the masses of the system, moving with a velocity equal to that of its centre of inertia, and of the motions of the separate parts relatively to the centre of inertia.

Let OI represent the velocity of the centre of inertia, IP that of any point of the system relative to 0. Then the actual velocity of that point is OP, and the proof of § 196 applies at once- -it being remembered that the mean of IQ, i. e. the mean of the velocities relative to the centre of inertia and parallel to OI, is zero by § 65.

235. The kinetic energy of rotation of a rigid system about any axis is (§§ 55, 179) expressed by mr2w2, where m is the mass of any part, r its distance from the axis, and w the angular velocity of rotation. It may evidently be written in the form w2mr2. The factor Emr2 is of course (§ 198) the Moment of Inertia of the system about the axis in question.

It is worth while to notice that the moment of momentum of any rigid system about an axis, being Emvr=wmr2, is the product of the angular velocity into the moment of inertia; while, as above, the half product of the moment of inertia by the square of the angular velocity is the kinetic energy.

If we take a quantity k, such that

k2Σm=Σmr2,

k is called the Radius of Gyration about the axis from which r is measured. The radius of gyration about any axis is therefore the distance from that axis at which, if the whole mass were placed, it I would have the same moment of inertia as before. In a fly-wheel, where it is desirable to have as great a moment of inertia with as

small a mass as possible, within certain limits of dimensions, the greater part of the mass is formed into a ring of the largest admissible diameter, and the radius of this ring is then approximately the radius of gyration of the whole.

236. The rate of increase of moment of momentum is thus, in Newton's notation (§ 28), wΣmr2; and, in the case of a body free to rotate about a fixed axis, is equal to the moment of the couple about that axis. Hence a constant couple gives uniform acceleration of angular Couple By § 178 we see that the corresponding Mk2

velocity; or =

237. For every rigid body there may be described about any point as centre, an ellipsoid (called Poinsot's Momental Ellipsoid) which is such that the length of any radius-vector is inversely proportional to the radius of gyration of the body about that radius-vector as axis.

The axes of the ellipsoid are the Principal Axes of inertia of the body at the point in question.

When the moments of inertia about two of these are equal, the ellipsoid becomes a spheroid, and the radius of gyration is the same for every axis in the plane of its equator.

When all three principal moments are equal, the ellipsoid becomes a sphere, and every axis has the same radius of gyration.

238. The principal axes at any point of a rigid body are normals to the three surfaces of the second order which pass through that point, and are confocal with an ellipsoid, having its centre at the centre of inertia, and its three principal diameters coincident with the three principal axes through these points, and equal respectively to the doubles of the radii of gyration round them. This ellipsoid is called the Central Ellipsoid.

239. A rigid body is said to be kinetically symmetrical about its centre of inertia when its moments of inertia about three principal axes through that point are equal; and therefore necessarily the moments of inertia about all axes through that point equal (§ 237), and all these axes principal axes. About it uniform spheres, cubes, and in general any complete crystalline solid of the first system (see chapter on Properties of Matter) are kinetically symmetrical.

A rigid body is kinetically symmetrical about an axis when this axis is one of the principal axes through the centre of inertia, and the moments of inertia about the other two, and therefore about any line in their plane, are equal. A spheroid, a square or equilateral triangular prism or plate, a circular ring, disc, or cylinder, or any complete crystal of the second or fourth system, is kinetically symmetrical about its axis.

240. The foundation of the abstract theory of energy is laid by Newton in an admirably distinct and compact manner in the sentence of his scholium already quoted (§ 229), in which he points out its