SCIENCE 



NEW YOEK, AUGUST 5, 1892. 



ON THE FUNDAMENTAL HYPOTHESES OF AB- 

 STRACT DYNAMICS.' 



BY PROFESSOR J. G. MACGREGOR, D.SC. 



The formally recognized axioms of abstract dynamics 

 employed by most writers are the three Laws of Motion 

 enunciated by Newton, in the " Principia," not always in the 

 form given them by Newton, but in some form or other. 

 It is obviously important that sucli axioms should be precise 

 in their enunciation, independent of one another, sufficient 

 for the deduction of all propositions applicable to natural 

 forces generally, and as few as possible. 



These axioms are sometimes regarded as constituting a 

 definition^ of force. As defining force, however, they are 

 not consistent with one another; for momentum being a 

 relative conception, i.e., having magnitude and direction 

 which vary with the point by reference to which velocity is 

 specified, force, if defined by the first and second laws, must 

 also be a relative conception. And it follows that the third 

 law cannot in general hold; for it is easy to show that if it 

 hold for one point of reference, it cannot hold for another 

 having an acceleration relative to the first. 



The axioms are thus statements about the action of force, 

 force being assumed to be already a familiar conception. As 

 applicable to the translation of bodies, they may be regarded 

 either as hypotheses verified by the deductions made from 

 them, or as generalizations established by direct though 

 rough experiments. When, however, we come to study the 

 effect of force in changing the rotation of bodies or their 

 state of strain, we assume the laws of motion to hold for the 

 small parts (particles or elements) of which we imagine the 

 bodies to consist. And therefore, as forming the basis of 

 dynamics as a whole, they must be regarded as hypotheses. 

 In either case it is necessary to note that both the popular 

 and the scientific conceptions of force ascribe to it a magni- 

 tude and direction quite independent of the point of reference 

 which may be used in specifying the motion of the body on 

 which it acts. 



I. The Precision of the Laws of Motion. 



Owing to this non-relative character of force, it is obvious 

 that the first and second laws of motion can hold only pro- 

 vided the motion of bodies be specified relatively to certain 

 points. In omitting the mention of these points, Newton's 

 laws are somewhat lacking in precision; and it is important 

 to determine what the points are. 



As, according to the first law, two particles which are both 

 free from the action of force must have uniform velocities, 

 relatively to the unspecified point of reference, each must 

 have a uniform velocity relatively to the other. Hence the 

 first law, as pointed out by Tait,^ holds relatively to any 

 particle on which no forces act. 



' Abstract of the presidential address to the Jlathematical and Physical 

 Section of th© Royal Society of Canada, at the meeting held May, 189"2. 

 ' Maxwell's Matter and Motion, Art. xl. 

 3 Properties of Matter (1S85), p. 92. 



As, according to the second law, the acceleration of 

 either a particle of finite mass acted upon by no force, 

 or a particle of infinite mass acted upon by no infinite 

 force, must be zero relatively to the unspecified point 

 of reference, this law must hold relatively to all such 

 particles. 



But such particles are fictitious. To bring the second law 

 within the region of practical application, we must find 

 accessible points by reference to which it holds. This may 

 readily be done ; for it is easy to prove it to hold for 

 a particle acted upon by given forces, relatively to any 

 other particle, with respect to which, but for the action 

 of these forces, the former would have no acceleration. 

 Thus, as is usually assumed, the acceleration, relative to 

 a point of the earth's surface, of a body situated at that 

 point and at rest or in uniform motion relatively to it, 

 except in so far as its motion may be modified by given 

 forces, may be determined by the application of the second 

 law. 



It is interesting to note that this was the point of reference 

 • employed by Newton in the experiments made by him to 

 verify the third law. In these well-known experiments^ on 

 the impact of spheres, the spheres were suspended by strings, 

 and impact was made to occur when the spheres occupied 

 their lowest positions. 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 verti- 

 cal; and the point of reference was consequently 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 acceleration relatively to this 

 point. Hence the equal and opposite changes of momentum 

 observed were specified by reference to a point with respect 

 to which, apart from the action of the stress due to impact, 

 the impinging spheres had no acceleration. 



As the third law asserts merely the equality and opposition 

 of two forces, it must hold for all points of reference; or 

 rather it is independent of points of reference. 



It follows that besides the points mentioned above, with 

 respect to whicli the second law holds, there is, in the case 

 of a system of particles, free from the action of external 

 force, another, viz., the centre of mass of the system. For 

 this point may be shown by the aid of the third law to have 

 no acceleration relatively to any point, by reference to whicli 

 the second law holds. 



It may easily be proved that the stress between two parti- 

 cles is proportional to the product, by the sum of their masses 

 into their relative acceleration ; and that consequently, if one 

 of the particles be of infinite mass, the stress is proportional 

 to the mass of the other multiplied by the relative accelera- 

 tion. Hence if, in applying the second law of motion, a 

 particle of infinite mass be chosen as point of reference, all 

 the forces acting on a system of particles, both external and 

 internal, may be regarded as exerted upon them by the 

 particle of infinite mass. 



' Principia: Scholium to Axiomata. 



