APPENDIX I TO CHAPTER II
NEWTON'S LAWS OF MOTION
Let () as in the accompanying figure be rectangular axes at rest; let () be the velocity of a material particle of mass at () relative to these axes, and let () be the acceleration of the same particle. Also let () be the force on the particle . The first two of Newton's laws can be compressed into the equations
It is unnecessary to trace the elementary consequences of these equations.
The third law of motion considers a fundamental characteristic of force and is founded on the sound principle that all agency is nothing else than relations between those entities which are among the ultimate data of science. The law is, Action and reaction are equal and opposite. This means that there must be particles ′, ″, ‴ etc. to whose agency () are due, and that we can write where () is due to ′ alone, () to ″ alone, and so on.
Furthermore let the particle ′ be at () and () be the acceleration of ′. Also let () be the force on ′; and let , etc. have meanings for ′ analogous to those which , etc. have for . Then according to the third law the two forces are equal and opposite, namely they are equal in magnitude, opposite in direction, and along the line joining and '. These requirements issue in two sets of equations with two analogous equations.
The two equal and opposite forces on and ′ due to their mutual direct agency, namely, together constitute what is called a 'stress between and '.
Thus the third law of motion falls into three parts, symbolised by the three sets of equations (2), (3) and (4). The set (2) expresses that all force on matter is due to stresses between it and other matter; and sets (3) and (4) express the two fundamental characteristics of stresses. We need not stop to enquire whether the short verbal expression of the law logically expresses these three properties. This is a minor point of exposition dependent on the context in which this formulation of the law is found.