The discussion of cases of equilibrium forms the subject of the science of Statics.

It is manifest that since the system of forces is exactly balanced, and is equivalent to no force at all, the forces will also be balanced if they act in the same way on any other material system, whatever be the mass of that system. This is the reason why the consideration of mass does not enter into statical investigations.

ARTICLE LIV.-THE THIRD LAW OF MOTION.

Law III.-Reaction is always equal and opposite to action, that is to say, the actions of two bodies upon each other are always equal and in opposite directions.

When the bodies between which the action takes place are not acted on by any other force, the changes in their respective momenta produced by the action are equal and in opposite directions.

The changes in the velocities of the two bodies are also in opposite directions, but not equal, except in the case of equal masses. In other cases the changes of velocity are in the inverse ratio of the masses.

ARTICLE LV.-ACTION AND REACTION ARE THE PARTIAL ASPECTS OF A STRESS.

We have already (Article XXXIII.) used the word Stress to denote the mutual action between two portions of matter. This word was borrowed from common language, and invested with a precise scientific meaning by the late Professor Rankine, to whom we are indebted for several other valuable scientific terms.

As soon as we have formed for ourselves the idea of a stress, such as the Tension of a rope or the Pressure between two bodies, and have recognised its double aspect as it affects the two portions of matter between which it acts, the third law of motion is seen to be equivalent to the statement that all force is of the nature of stress, that stress exists only between two portions

ACTION AND REACTION.

47

of matter, and that its effects on these portions of matter (measured by the momentum generated in a given time) are equal and opposite.

The stress is measured numerically by the force exerted on either of the two portions of matter. It is distinguished as a tension when the force acting on either portion is towards the other, and as a pressure when the force acting on either portion is away from the other.

When the force is inclined to the surface which separates the two portions of matter the stress cannot be distinguished by any term in ordinary language, but must be defined by technical mathematical terms.

When a tension is exerted between two bodies by the medium of a string, the stress, properly speaking, is between any two parts into which the string may be supposed to be divided by an imaginary section or transverse interface. If, however, we neglect the weight of the string, each portion of the string is in equilibrium under the action of the tensions at its extremities, so that the tensions at any two transverse interfaces of the string must be the same. For this reason we often speak of the tension of the string as a whole, without specifying any particular section of it, and also the tension between the two bodies, without considering the nature of the string through which the tension is exerted.

ARTICLE LVI.—ATTRACTION AND REPULSION.

There are other cases in which two bodies at a distance appear mutually to act on each other, though we are not able to detect any intermediate body, like the string in the former example, through which the action takes place. For instance, two magnets or two electrified bodies appear to act on each other when placed at considerable distances apart, and the motions of the heavenly bodies are observed to be affected in a manner which depends on their relative position.

This mutual action between distant bodies is called attraction when it tends to bring them nearer, and repulsion when it tends to separate them.

In all cases, however, the action and reaction between the bodies are equal and opposite.

ARTICLE LVII.-THE THIRD LAW TRUE OF ACTION AT A DISTANCE.

The fact that a magnet draws iron towards it was noticed by the ancients, but no attention was paid to the force with which the iron attracts the magnet. Newton, however, by placing the magnet in one vessel and the iron in another, and floating both vessels in water so as to touch each other, showed experimentally that as neither vessel was able to propel the other along with itself through the water, the attraction of the iron on the magnet must be equal and opposite to that of the magnet on the iron, both being equal to the pressure between the two vessels.

Having given this experimental illustration Newton goes on to point out the consequence of denying the truth of this law. For instance, if the attraction of any part of the earth, say a mountain, upon the remainder of the earth were greater or less than that of the remainder of the earth upon the mountain, there would be a residual force, acting upon the system of the earth and the mountain as a whole, which would cause it to move off, with an ever-increasing velocity, through infinite space.

ARTICLE LVIII.-NEWTON'S PROOF NOT EXPERIMENTAL.

This is contrary to the first law of motion, which asserts that a body does not change its state of motion unless acted on by external force. It cannot be affirmed to be contrary to experience, for the effect of an inequality between the attraction of the earth on the mountain and the mountain on the earth would be the

same as that of a force equal to the difference of these attractions acting in the direction of the line joining the centre of the earth with the mountain.

If the mountain were at the equator the earth would be made to rotate about an axis parallel to the axis about which it would otherwise rotate, but not passing exactly through the centre of the earth's mass.

If the mountain were at one of the poles, the constant force parallel to the earth's axis would cause the orbit of the earth about the sun to be slightly shifted to the north or south of a plane passing through the centre of the sun's mass.

If the mountain were at any other part of the earth's surface its effect would be partly of the one kind and partly of the other.

Neither of these effects, unless they were very large, could be detected by direct astronomical observations, and the indirect method of detecting small forces, by their effect in slowly altering the elements of a planet's orbit, presupposes that the law of gravitation is known to be true. To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus.

We cannot, therefore, regard Newton's statement as an appeal to experience and observation, but rather as a deduction of the third law of motion from the first.

CHAPTER IV.

ON THE PROPERTIES OF THE CENTRE OF MASS OF A MATERIAL SYSTEM,

ARTICLE LIX.-DEFINITION OF A MASS-VECTOR.

WE have seen that a vector represents the operation of carrying a tracing point from a given origin to a given point.

Let us define a mass-vector as the operation of carrying a given mass from the origin to the given point. The direction of the mass-vector is the same as that of the vector of the mass, but its magnitude is the product of the mass into the vector of the mass.

Thus if OA is the vector of the mass A, the massvector is O AA.

ARTICLE LX.-CENTRE OF MASS OF TWO PARTICLES.

If A and B are two masses, and if a point C be taken in the straight line A B, so that B C is to CA as A to B, then the mass-vector of a mass A+B placed at C is equal to the sum of the mass-vectors of A and B. For O AA+0 BB (OC+C A) A+ (Ö C+C B)B. =OC(A+B) +CAA+CB.B.

Fig. 7.

A

=

Now the mass-vectors CA⚫A and C B B are equal and opposite, and so destroy each other, so that O A A+ OBB OC (A+B)

or, C is a point such that if the masses of A and B were concentrated at C, their mass-vector from any origin O would be the same when A and B are in their actual positions. The point C is called the Centre of Mass of A and B.

ARTICLE LXI.-CENTRE OF MASS OF A SYSTEM.

as

If the system consists of any number of particles, we may begin by finding the centre of mass of any two particles, and substituting for the two particles a particle equal to their sum placed at their centre of mass. We may then find the centre of mass of this particle, together with the third particle of the system, and place the sum of the three particles at this point, and so on