6 J. G. MacGEEGOE on THE 



same value as it has relatively to the unspecified point and axes. Novs^ it follows from 

 this law that a particle free from the action of force will have no acceleration relatively 

 to the unspecified point and axes of referenci>, and that lines drawn from it to other par- 

 ticles unacted on by force and having the same velocity, will have a constant inclination 

 to the unspecified axes. Hence, relatively to this point and these axes of reference an 

 acceleration will have the same value as it has relatively to the unspecified point and 

 axes. And thus the second law, like the first, holds relatively to any set of particles on 

 which no forces act and which haA'^e the same velocity. Its enunciation may thus be 

 made precise by a modification similar to that suggested in the case of the first law. 



As it follows from the second law that a particle of infinite mass, acted upon by no 

 infinite force, has zero acceleration, it is obvious that the second law, and therefore 

 the first as well, hold also by reference to axes determined by any set of particles of in- 

 finite mass acted upon by no infinite force and having the same velocity. 



But both finite particles on which no forces act, and particles of infinite mass on 

 which no infinite forces act, are fictitious. To bring these laws within the region of prac- 

 tical application, therefore, we must find accessible points and axes by reference to which 

 they hold. And this may readily be done. For if A, B, C and D form a set of particles by 

 reference to which (i.e. with A, say, as point of reference and AB, AC, AD, as axes of refer- 

 ence) the acceleration of a body E is equal to the quotient of the force acting on it by its 

 mass, and if we act on all these bodies with such forces as will produce in them all equal 

 accelerations in the same direction relatively to any point and axes, the acceleration of E 

 relative to A will be the same as before. Thus if any set of particles be so acted upon by 

 forces as to have all the same acceleration, all or all but one having the same velocity, 

 relatively to any point and axes, and if an additional force or additional forces act upon 

 this one, the acceleration produced or the component accelerations produced, if specified 

 relatively to the other particles, will be proportional to the quotient of the force, or the 

 quotients of the forces, by the mass of the particle. This is what is always assumed in 

 dealing with the motions of bodies on the earth's surface. Neighbouring points on the 

 earth's surface and bodies situated near them, whether they be at rest or in uniform 

 motion relatively to them, have all practically the same acceleration. Hence the accelera- 

 tions of such bodies relatively to these points may be determined by the application of the 

 second law, and the absence of such acceleration in them may betaken, in accordance with 

 the first law, as sufficient evidence that no force is acting on them save that which gives 

 them the acceleration which they have in common with the neighbouring points of the 

 earth's surface. Even in cases in which the forces acting on bodies at the earth's surface 

 are stresses between the bodies and the earth itself, as in the great majority of cases in 

 practice, the second law may be considered to hold relatively to points of the earth's sur- 

 face, without any sensible error. For, owing to the relatively enormous mass of the earth, 

 the accelerations produced by such stresses in the points of reference themselves are so 

 small, compared with the accelerations produced in the bodies, as to be practically 

 negligible. 



It is interesting to note that it was a point on the earth's surface, that was employed 

 by Newton as point of reference in the experiments made by him to verify the third law.' 



1 Principia : Scholium to Axiomata. 



