242 Prof. J. Gr. MacGregor on Abstract Dynamics and 



results in the production of an equivalent amount of potential 

 energy. This is equivalent to the assumption that the above 

 expression is a complete differential, which is again equiva- 

 lent to the assumption that the stress components, P, Q, R, 

 S, T, U, at a point, are proportional to the rates of change, 

 with respect to the corresponding strain components, e,f, g, 

 a, b, c respectively, of a function of all these strain com- 

 ponents. 



The Third Law and the hypothesis just enunciated are both 

 statements partially specifying natural stresses. We may 

 combine them in one by assuming that natural forces may be 

 regarded as stresses between contiguous elements of a body 

 (or medium), the components of the stress at a point having 

 the relations as to magnitude just specified. 



Thus in cases of contact-action also, the purely dynamical 

 hypotheses reduce to two, — (1) The Law of Force — 

 Newton's Second Law, and (2) the Law of Stress, as just 

 enunciated. In such cases also there is, however, a third 

 hypothesis, viz., (3) the Law of the constitution* of bodies, 

 that bodies may be regarded as consisting of elements exerting 

 forces upon contiguous elements only, across their surfaces 

 of contact. 



The above results have a bearing on the controversy with 

 regard to the rari-constant and the multi-constant theories of 

 elasticity. For in order to form an estimate of the relative 

 probability of deductions from the two theories, accuracy in 

 deduction being assumed, we must compare the hypotheses 

 employed. 



The multi-constant theorists, in applying the contact-action 

 conception of bodies, have usually employed as dynamical 

 hypotheses the Second and Third Laws of Motion and the 

 Law of the conservation of energy, which together are equi- 

 valent in hypothetical content to the above Laws of Force and 

 of Stress, or to the Laws of the conservation and the trans- 

 ference of energy. 



The rari-constant theorists have used the molecular, or 

 rather the point-atom, conception of bodies, and have em- 

 ployed as dynamical hypotheses the Second Law and the 

 assumption that the stress between any pair of particles is a 

 function of their distance, not of the distances of all the pairs 

 of particles of the system. Their dynamical hypotheses have 

 thus a greater hypothetical content than the Laws of Force 

 and Stress, and therefore also than the Laws of Energy. 



It would appear, however, that the discrepancy between 

 the results deduced from the two theories with regard to the 

 number of the elastic constants is not due to the additional 



