18 J. G. MacGEEGOE on THE 



has logical advantages to compensate for its educational drawbacks, it would seem to be 

 better to select as an additional law of motion one which will admit of the retention of 

 Newton's time-honoured laws. 



"When the impossibility of the perpetual motion is employed as an axiom it is used 

 merely as a means of showing that natural forces are conservative, and this proposition 

 may be deduced from it ' without the employment of any other hypothesis. So far as 

 their hypothetical content is concerned, therefore, the impossibility of the perpetual motion 

 and the conservatism of natural forces are identical. The only advantage the former would 

 appear to have over the latter, as an axiom, is that it is in closer touch with exj)erience. 

 Though men have sought out many inventions they have found the perpetual motion 

 impossible of attainment, while experience cannot be said to have shown in any direct 

 way that natural forces are conservative. This would doubtless be an advantage if, in 

 passing from the one to the other of the above propositions, we could keep in touch with 

 experience. That we cannot do so, however, is shown by any of the processes by 

 which the passage is made. Thomson and Tait make it in this way : " If more work is 

 done by the mutual forces on the different parts of the system in passing from one particular 

 configuration to another, by one set of paths than by another set of paths, let the system' 

 be directed by frictiouless constraint to pass from the first configuration to the second by 

 one set of paths and return by the other, over and over again. It will be a continual 

 source of energy without any consumption of materials, which is impossible; "and it is 

 obvious that when we employ frictiouless constraint we lose our touch with experience. 

 There would thus seem to be no advantage in the impossibility of the perpetual motion 

 over the conservatism of natural forces, as an axiom. 



The hypothesis of the conservatism of natural forces is independent of the second law 

 of motion, but is closely allied to the third. This latter states that the action and reaction 

 between any two particles of a system are equal and opposite, and in applying it, is 

 always interpreted as stating also that they are in the straight line joining the particles. 

 The hypothesis of conservatism states that if X, Y, Z are the rectangular components 

 of the resultant of all the forces acting on any particle of a system, and if ^.-c, dy, dz are the 

 components of its displacement during any small change of configuration, '2 (Xdx -\- Ydy 

 + Zdz) must be a complete differential of a function of the co-ordinates of the particles 

 of the system, and, therefore, that the magnitude and direction of the resultant of these 

 forces on each particle must depend upon the configuration of the system and upon nothing 

 else. 



Both the third law and the hypothesis of conservatism thus tell us .something about 

 the stresses between the particles of a system. If we combine them we obtain the sum 

 total of the information given by both together, viz.. that since, by the third law, action 

 and reaction are equal, opposite and in the line joining the particles, if S is the stress be- 

 tween any two particles and s the distance between the particles, we have 2b' {Xdx + Yd// + 

 Zdz)=2 Sds, and since by the hypothesis of conservatism this must be a complete diflerential, 

 each stress may depend upon the distances of the various pairs of particles in the system 

 and can depend upon nothing else. In the particular case of a system of two particles, 

 the two axioms together tell us that the stress between them must be in the line joining 



• See Thomson and Tait: Treatise on Natural Philosophy, Vol. I. Part 1 (1879), U 271, 272. 



