248 Prof. J. G. MacGregor on the 



others acting, and that, therefore, in the case of all the 

 stresses action and reaction are equal and opposite. 



It is at once obvious that this argument will hold also, 

 whether we restrict ourselves to rigid systems or not. Prof. 

 Lodge does so, probably because he feels he cannot appeal to 

 the experience of the human race with regard to the motion 

 of the centre of mass of a non-rigid system. Had he adopted 

 as his axiom the generalized first law (which he would be 

 justified in doing according to my conception of an axiom as 

 a proposition by means of which it is found possible to co- 

 ordinate dynamical phenomena generally), then, with the aid 

 of the second law and the physical independence of stresses, 

 he might have deduced the equality and opposition of the 

 action and reaction of all stresses *. But even then he would 

 have deduced only what is explicitly stated in the third law 

 and not the whole law. For just as the second law, by the 

 generality of its assertion, implies the "physical indepen- 

 dence of forces," so the third law implies the physical 

 independence of stresses, at least so far as the equality 

 and opposition of their action and reaction are concerned. 

 This implied part of the third law is assumed in the above 

 deduction. 



So much for the asserted possibility of deducing the third 

 law from the first. Prof. Lodge has held also that it may 

 be deduced from the second f . Divested of its " muscles and 

 clothing," his argument is as follows : — Action maybe taken 

 to mean simply the whole force applied to the body con- 

 sidered. The reaction of a body is defined as equal to the 

 product of its mass into its acceleration. The second law of 

 motion may be expressed in an equation on the one side of 

 which we have the resultant force on a body or the action, 

 on the other side of which we have its mass multiplied by its 

 acceleration, which we have agreed to call its reaction. Thus 

 action is equal to reaction. After a few paragraphs of 

 explanatory matter he changes the expression of this result, 

 without any attempt at justification, to the following : — The 

 reaction or mass acceleration of a body is equal and opposite 

 to the resultant of all the forces acting on it. It is hardly 

 necessary to discuss this argument. It will be sufficiently 

 obvious that if the definitions of action and reaction be ac- 

 cepted, reaction, if assumed to have direction, must be 

 co-directional with action, not opposite to it, and that there- 



* Streintz (loc. cit. p. 131) and Muirhead (loe. cit. p. 477) point out 

 the possibility of deducing the third law from the generalized first law, 

 but do not perceive the necessity of assuming the physical independence 

 of stresses. 



f 'The Engineer,' vol. lix. (1885), pp. 217, 311, 380. 



