12 J. G. MacGEEGOR on THE 



servation of the motion of the centre of mass of a system of bodies. It is with this 

 deduction from the first law, therefore, that the acceleration of the system is asserted to 

 be inconsistent. But in making this deduction he had assumed the third law. If, 

 therefore, in making it he regarded this law as directly applicable only to contact actions, 

 the deduction could only apply to systems exerting contact actions, and the acceleration 

 of a system of attracting bodies, unacted upon by external force, would not be incon- 

 sistent with it. If, on the other hand, in making the deduction he regarded the third 

 law as applicable to all stresses, his application of the deduction in the argument under 

 consideration would be a begging of the question. 



That Newton should have tripped in so simple an argument is of course possible, 

 but is highly improbable. It is much more probable that the discussion which is 

 generally regarded as a deduction of the third law of motion from the first, but which I 

 think I have shown cannot be regarded as having been meant to be more than a deduc- 

 tion of one part of the third law from the other part and from the first, is in reality 

 merely an illustrative comment. 



Maxwell, as I have said, regards Newton's argument " as a deduction of the third law 

 of motion from the first." He paraphrases it as follows' : — " If the attraction of any part 

 of the earth, say, a mountain, upon the remainder of the earth, were greater or less 

 than that of the remainder of the earth upon the mountain, there would be a residual 

 force acting upon the system of the earth and the mountain as a whole, which would 

 cause it to move off with an ever-increasing velocity through infinite space. This is con- 

 trary to the first law of motion, which asserts that a body does not change its state of 

 motion unless acted upon by external force." Whether or not I am right in thinking 

 that Maxwell misrepresents Newton in the view he takes of his argument, he himself 

 obviously considered the above paraphrase of it to be a sound deduction. It may, how- 

 ever, be attacked on two grounds. First, it assumes that the attraction between the 

 mountain and the remainder of the earth is the only stress between them, ignoring the 

 stress at their surface of contact, an inequality in the action and reaction of which might 

 obviously neutralise the " residual force " due to the assumed inequality in the action 

 and reaction of the attraction. And secondly, the conclusion drawn is not the third law 

 of motion. For the third law asserts the equality and opposition of the action and 

 reaction between two bodies, not between two parts of one body. That this criticism is 

 sound becomes especially obvious when we reflect that the laws of motion, as fundamen- 

 tal axioms of dynamics, must be applicable to particles, not to extended bodies. It is 

 clear that the above argument is not applicable in the case of a particle ; and it is also 

 clear that if the laws of motion be enunciated as applicable directly to particles, the 

 conservation of the motion of the centre of mass of an extended system cannot be 

 established without the assumption of the third law. 



"We conclude then with regard to the independence of the laws of motion that while 

 the first law is re-enunciated in the second, the second and third have not been proved to 

 be capable of deduction either one from the other or from more simple hypotheses. 



"With regard to the sufficiency of the laws of motion to give by deduction all the laws 

 of abstract dynamics, the best test to apply is the question : — Can we deduce from them 



' Matter and Motion, Arts. Ivii. and Iviii. 



