182 The Eev. J. H Jellett on the Equilibrium and Motion of an Elastic Solid. 



We shall now proceed to investigate the equations of equilibrium and mo- 

 tion, for bodies of the first class. 



I. — Hypothesis of Independent Action. 



4. Let the several particles of a body, which satisfies the hypothesis of inde- 

 pendent action, be displaced from their original position of free equilibrium, 

 this displacement being supposed to follow some regular law. Let it be re- 

 quired to determine the conditions of equilibrium of these jiarticles in their 

 new position, or, in other words, to assign the forces which should be applied 

 to each of them in order to keep them at rest. Again, if the particles be left 

 to themselves after the displacement, let it be required to determine the law of 

 their motion. 



In applying the method of Lagrange to any problem of equilibrium or 

 motion, it is plainly necessary to commence with two assumptions, namely: — 

 L An assumed expression for the intensity of each of the acting forces. 2. An 

 assumed expression for the effect which this force tends to produce; the effect 

 of a force being defined by the quantity which it tends to change. 



Let TO, m' be two particles of the body under consideration, and let F be 

 the force which, in their displaced position, they exert upon each other. 



Let X, y, z be the co-ordinates of m in its original position, and |, ?/, f its 

 resolved displacements. Let also x\ y', z', ^', r/, f' be the co-ordinates and 

 displacements of m' .Then, since, by the hjrpothesis of independent action, F 

 does not depend upon the displacement of any of the other particles, and since, 

 if the body have a regular constitution, the state of each particle must be a 

 function of its position, 



F=f {x, y, z, x', y', z\ ^, n, T' ^. '/. ^); 

 or, as it may be otherwise written, 



F=fix,y,z, x\y',z', I rj, ^, ^'-?, v' - V, ^' - O- 



But in all media with which we' are acquainted, no internal force appears 

 to be generated by a mere transference of the entire system from one position 

 in space to another, the relative positions of the several particles remaining un- 

 changed. 



