the question of the number of the Elastic Constants, 241 



Law of Force — Newton's Second Law, and (2) the Law of 

 Stress, just enunciated. To these, however, must be added 

 a third, viz., (3) the Law of the constitution of bodies, that 

 bodies may be regarded as consisting of particles acting on 

 one another at a distance. 



In the paper * of which this is an abstract, a similar attempt 

 is made to formulate the hypotheses employed when, as in 

 the study of fluids and elastic solids, bodies are considered as 

 consisting of elements which exert forces on contiguous 

 elements only, across surfaces of contact. In obtaining the 

 equations of motion, in this case, the Third Law is applied, 

 when, the traction at x, y, z on one end of a parallelopiped 

 with dx, dy, dz as edges being called P, the traction on the 



other end is put equal to — ( P + ^— dx J. The Second Law 



is partially applied when the quotient of the force on an 

 element by its acceleration is put equal to p dx dy dz, p being 

 the density. It is only partially applied, however ; for as 

 p varies with the time, there is nothing in the resulting 

 equation to show that the quotient of force by acceleration is 

 constant, as the Second Law states. Accordingly the equa- 

 tions of motion thus obtained are insufficient completely to 

 determine the motion. An additional equation is necessary, 

 viz., one which completes the application of the Second Law 

 by expressing in some form or other, that the p dx dy dz of 

 the equations of motion is constant. This is the so-called 

 equation of continuity, which is thus only a partial application 

 of the Second Law. It was regarded by Rankine as re- 

 quiring an independent axiom t> and is derived by other 

 writers by asserting, in a vague kind of way, the impossibility 

 of the annihilation and the creation of matter, the constancy 

 of mass, or the continuity of the motion considered. 



In order to obtain the law of the conservation of energy, 

 it is necessary to assume, in addition, that the work done J 

 by the stress components during a strain, viz., the integral, 

 between the initial and final states of strain, of 



JIfCP 



de + Qdf+ Rdg + Sda + Tdb + JJdc) dx dy dz, 



* Trans. Roy. Soc. Canada [2], vol. i. sec. iii. p. 85 (1895). 



t ' Applied Mechanics,' 9th eel. p. 411. 



| I use the term— work done by a force — in its ordinary sense, as 

 being the product of the force into the component, in the direction of 

 the force, of the displacement (relative, of course, to a dynamical refer- 

 ence system) of its point or place of application. The definition of this 

 term which Newcomb (Phil. Mag-. [5] xxvii. (1889) p. 115) proposed to 

 substitute for the ordinary one would not be suited to the contact-action 

 conception. See note in the Proc. and Trans, of the Nova Scotian 

 Institute of Science, vol. viii. ( 1890-94) p. 4G0. 



Phil Mag. S. 5. Vol. 42. No. 256. Sept. 1896. T 



