Hypotheses of Dynamics. 247 



held to depend on how general the terms are in which that 

 law has been stated. If it can be axiomatically asserted that 

 the centre of mass of a rigid system moves uniformly until an 

 external force acts on the system, and also that the system 

 does not begin to spin, then the third law is established. For 

 since zero acceleration means zero force, it follows that all 

 the internal forces add up to zero, and have no moment ; and 

 since the system can be dissected bit by bit without ceasing 

 to be a system within the scope of the first law, it follows 

 that no stress can contain an unbalanced force or couple. " 

 Here, then, we have a new deduction, on which I would make 

 two remarks : — (1) The conclusion is obviously too general. 

 For since the assumptions specified are made for a rigid 

 system only, the " no stress " of the conclusion should clearly 

 be — no stress between the parts of a rigid system. The con- 

 clusion would thus become only a particular case of the third 

 law. (2) That even this modified conclusion cannot be ob- 

 tained without additional assumptions, and, even with them, 

 by the method of dissection, may readily be shown. " All 

 the internal forces add up to zero and have no moment." 

 How do we know this? Only by the aid of familiar deduc- 

 tions from the second law of motion. Thus the second law 

 is assumed. Dissect away one particle from the system. 

 By the second law, as above, the internal forces of the re- 

 maining particles now add up to zero and have no moment. 

 But we cannot assert this to have been true before the 

 removal of the particle, unless we assume the physical inde- 

 pendence of stresses. If this second additional assumption 

 be made, though we now know that the actions and reactions 

 of the stresses between any one particle and the remaining 

 particles add up to zero and have no moment, we cannot con- 

 clude that "no stress can contain an unbalanced force or 

 couple," because they would add up to zero and would have 

 no moment, also, provided any inequality in the action and 

 reaction of one stress, and their resultant moment, were neu- 

 tralized by inequalities in the actions and reactions of other 

 stresses and by their resultant moment, respectively. 



While Prof. Lodge's method of dissection will not give 

 even the modified conclusion, even with the aid of the above 

 additional axioms, the reverse process will give it without 

 his assumption as to spin. For in a system of two particles 

 the conservation of motion of the centre of mass and the 

 second law together tell us that the action and reaction of the 

 single stress in the system are equal and opposite. And in 

 a system of any number of particles, the axiom of the 

 physical independence of stresses tells us that the stress 

 between any two particles is the same as if there were no 



