20 J. G. MaoGEEGOR ON THE 



which he deduces the law of the couservation of euergy are " identical," and that " in 

 systems to which the principle of the conservation of force can be applied, in all its gene- 

 rality, the elementary forces of the material points must be central forces ; " and he 

 considers himself to have proved this statement by showing that the force must be central 

 in a system consisting of two particles. Williamson and Tarleton ' quote his i)roof with- 

 out protest against the assumption involved in it. Mr. W. E. Browne in a paper- in- 

 tended to prove that the law of the conservation of energy and the axiom that natural 

 forces are central forces, " imply each other," confines himself to the case of two particles, 

 and Mr. Gr. W. vOn Tunzelmaun, ' in criticising the paper, makes no objection to it on 

 this ground. The additional hypothesis is therefore seen to be already recognized as 

 axiomatic, though it has never, so far as I am aware, been formulated as a law of motion. 

 Since then the adoption, as a law of motion, of the hypothesis, that natural forces may 

 be regarded as central forces, would admit of the retention of Newton's laws, and would 

 indeed make them a complete statement with regard to natural forces, since it is siifficient 

 for the deduction of the law of the conservation of energy, and since it includes no unre- 

 cognized hypothesis over and above what is necessary for this deduction, it would seem 

 to commend itself for enunciation as a law of motion, more than any of the other hypotheses 

 referred to above. 



In formulating this hypothesis as a law of motion it will be obvious that, as in the 

 case of the third law of motion, it is unnecessary to specify points and directions of re- 

 ference. For the hypothesis simply states the direction of a certain force to be the same 

 as that of a certain line and the magnitude of the force to be dependent upon the length 

 of that line. 



It may be well to note here an obvious corollary from this conclusion, viz., that as the 

 law of the conservation of energy may be derived from this hypothesis and the second law 

 of motion, and as the former holds for all points of reference, the law of conservation must 

 hold for all points by reference to which the second law holds. This conclusion is incon- 

 sistent with Newcomb's assertion ^ that this law " assumes that we refer the motions of all 

 the bodies whose energy is considered to some foreign body of infinite mass, from which 

 emanate the forces which give motion to the system." According to the above, this law 

 may of course be expressed relatively to a body of infinite mass, and, if thus expressed, the 

 forces which give motion to the system may be supposed to emanate from that body. But 

 it may also be expressed relatively either to a particle of finite mass free from the action of 

 force, or to the centre of mass of the system itself whose energy is conserved. 



The above assertion of Newcomb's occurs in a paper which ends as follows : — " I 

 cannot but think that sound philosophy would be promoted could these limitations of the 

 conservation of energy be made clear to those philosophers who see in the doctrine only 

 a special case of a general law of mind and matter." It may doubtless be well that the 

 attention of philosophers should be called to the limitations of the law of the conservation 

 of energy ; but the statement that it makes the assumption referred to above, gives it a 

 much more fictitious appearance than it really ought to present, and is thus misleading. 



' Dynamics (1885), p. 398. 



■^ Phil. Mag., Ser. 5, Vol. xv. (1883), p. 35. 



» Ibid., p. 152. 



* Phil. Mag., Ser. 5, Vol. xxvii. (1889), p. 116. 



