HYPOTHESES OP DYNAMICS. 19 



the particles and can depend only upon the distance between them. But they do not give 

 us the same complete information as to the magnitude of the stress between two particles 

 in a system consisting of more than two. Thus these two hypotheses together give us 

 only a partial statement of the law governing natural stresses. 



While, therefore, the enunciation of the hypothesis of the conservatism of natural 

 forces, would admit of the retention of Newton's laws and would form an extension of 

 them, the lav^s of motion thus reinforced, would still be incomplete and thus logically de- 

 fective, though of course they might possibly be as complete as any hypotheses warranted 

 by the present state of dynamical science. 



The second of the hypotheses mentioned, that natural forces may be regarded as 

 central forces, is, like that just considered, independent of the second law of motion and 

 closely allied to the third. But the alliance is so close in the case of this hypothesis that 

 the third law is but a partial statement of it. For when it is asserted that natural forces 

 may be regarded as attractions or repulsions it is implied that their action and reaction 

 are opposite and in the line joining the particles between which they act, and when it is 

 asserted that natural forces depend only on the distance of the particles between which 

 they act, it is implied that the action and reaction are equal. This hypothesis in fact is a 

 complete statement about the stresses acting between particles and thus includes the 

 incomplete statement of the third law. 



The hypothesis .under consideration is thus wider than the combined assumption ol 

 the conservatism of natural forces and of the third law of motion, and we must therefore 

 inquire whether or not the additional hypothesis involved in it is axiomatic. "What the 

 additional hypothesis is, is easily seen. For the conservatism of natural forces along with 

 the third law of motion tells us that, while in a system of two particles the stress depends 

 only on their distance, in a system consisting of more than two particles the stress between 

 any two may depend on the distances of the various pairs of particles in the system. The 

 axiom under consideration tells us that under all circumstances the stress between two 

 particles depends only on the distance between them. Thus the additional hypothesis 

 involved in assuming natural forces to be central forces over and above that involved in 

 assuming them to be both subject to the third law and conservative, is that the stress be- 

 tween two particles is not changed by the action of other stresses between them and 

 other particles. 



This additional hypothesis is one which it is customary to make in whole or in part, 

 in dealing with many dynamical problems. For example, in applying the third law of 

 motion, as already noted, to the discussion of rotation, it is, I think, invariably assumed 

 that the action and reaction of the stress between two particles are not only equal and 

 opposite, but also in the straight line joining the particles, or in other words, that the 

 direction of the stress between them depends only on the position of the two particles 

 concerned, not upon the configuration of the whole system. In the case of forces, such as 

 gravitational, electrical and magnetic attractions, whose laws have been determined, the 

 hypothesis under consideration may be said to have been proved to hold, so extensive is 

 the experimental verification which it has received. And in investigations into the laws 

 offerees whose laws are not yet known, it is, so far as my knowledge goes, invariably 

 made. So customary is it to make this assumption that it is usually made without com- 

 ment. Thus Helmholtz, in the memoir cited above, states that the two maxims from 



