Mr. R. F. Muirhead on the Laws of Motion. 477 



whose motion may be taken to represent that of the system, 

 which implies that the 3rd Law is true so far as the parts of 

 such a system are concerned. Now it seems difficult to draw 

 a valid distinction between such a system and any mass- 

 system whatever ; in fact it seems quite as legitimate to 

 assume that every mass-system has a centre of Inertia. 



But if this assumption were made, then clearly the first 

 Law could be deduced from the third in all its generality, 

 and vice versa. 



We see that in this respect again Newton ; s arrangement 

 is defective. We find that the experimental fact is not 

 stated directly, but implied in the assumption of the existence 

 of a mass-centre. In fact, strictly read, Newton's Definitions 

 and Axioms abound in logical circles, nugatory statements, 

 and illusory definitions ; and what real meaning they imply 

 is not at all explicit. 



The need for the removal of many obscurities which pertain 

 to the science of Dynamics as set forth in the Principia of 

 Newton, and in the writings of his successors, has been clearly 

 perceived by Professor James Thomson. In his paper on the 

 " Law of Inertia, &c," * he propounds the following Law of 

 Inertia : — 



" For any set of bodies acted on each by any force, a 

 Reference-Frame and a Reference Dial- traveller are kine- 

 matically possible, such that relatively to them conjointly the 

 motion of the mass-centre of each body undergoes change 

 simultaneously with any infinitely short element of the dial- 

 traveller progress, or with any element during which the 

 force on the body does not alter in direction nor in magnitude, 

 which change is proportional to the intensity of the force act- 

 ing on that body, and to the simultaneous progress of the 

 dial-traveller, and is made in the direction of the force." 



For explanations of the terms used I refer to the paper 

 itself. At the end of this paper we have the assertion : " The 

 Law of Inertia here enunciated sets forth all the truth which 

 is either explicitly stated, or is suggested by the First and 

 Second Laws in Sir Isaac Newton's arrangement." 



Professor Thomson's Law is doubtless, so far as order and 

 logic are concerned, an immense advance on the Newtonian 

 arrangement. Let us inquire whether it can be accepted as 

 absolutely satisfactory. 



How are we to measure the " forces w referred to ? If 

 kinematically, then we are again involved in a logical circle, 

 as may be seen by substituting in the Law, for the words 



* Proc. R. S. E. 1883-4, p. 568. 



