HYPOTHESES OF DYNAMICS. 7 



In these well-known experiments on the impact of spheres, the spheres were suspended 

 by strings, so attached that when the spheres were in their equilibrium positions, they 

 were in contact, their centres were in the same horizontal plane, and the strings were 

 vertical. Impact was made to occur when the spheres occvtpied their lowest positions, 

 one or both being drawn aside and let go. Their velocities before and after impact were 

 taken to be proportional to the chords of the arcs (corrected for resistance of air), through 

 which they had fallen, or were found to rise, respectively. Hence the acceleration of a 

 freely falling body was assumed to be vertical ; and the point of reference was conse- 

 quently the point of the earth's surface at which the experiments were made. Also at 

 the instant of impact, the spheres were passing through their positions of zero accelera- 

 tion relatively to this point. Hence the equal and opposite changes of momentum ob- 

 served were specified by reference io a point with respect to which, apart from the action 

 of the stress due to impact, the impinging spheres had no acceleration. 



Were we engaged in determining the relation of force to acceleration by direct experi- 

 ment, we would make our experiments necessarily on the earth's surface, and would find, 

 if we exerted no forces on a body, that the more completely we could eliminate the action 

 of such forces as friction, the more nearly would the body exhibit a uniform velocity rela- 

 tively to points near at hand on the earth's surface, and if we did exert forces upon it, that 

 the more completely we could eliminate friction, the more nearly would the quotient of 

 the resultant of these forces by the mass of the body represent the acceleration relative to 

 such points. Having thus found that the quotient of force by mass measures the acceler- 

 ation of a body relative to points which apart from the action of the force would have 

 the same acceleration as the body and which themselves have the same velocities, we could 

 argue backwards to the expression of the second law obtained above. 



The third law asserts merely the equality and opposition of two forces. If, therefore, 

 we regard force as a quantity which does not vary with the points of reference employed 

 in specifying motion, this law must hold whatever our points of reference may be. It is 

 independent of points of reference. 



The third law thus holding for all points of reference, it follows that the centre of 

 mass of a system of particles which is free from the action of external force, can have 

 no acceleration relatively to points by reference to which the second law holds. Hence 

 the centre of mass of such a system may be used as point of reference in applying the 

 second law. 



It may be well to notice here a fiction sometimes found useful in expressing dynami- 

 cal laws, viz., that in treating the motion of a system we may refer the motion to a par- 

 ticle or body of infinite mass, and regard the forces acting on the particles of the system 

 as exerted on them by the body of infinite mass, according to the third law of motion, the 

 third law being taken, if we use a particle of infinite mass as point of reference, as assert- 

 ing that action and reaction are equal and opposite, but not that they are in the same 

 straight line. That this fiction is permissible follows from the results noted above, that 

 a particle of infinite mass may be employed as point of reference in applying the second 

 law and that the third law holds for all points of reference, and from the consideration 

 that as no finite force can produce a change in the motion of a body of infinite mass, we 

 may imagine such a body to be acted upon by forces equal and opposite to the forces act- 



