The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 209 



111-4 cos^ a cos a dm + ll\Ai cos^ a, cos a/ dm,. 



But as tlie limits of integration are still variable, the form of the general equa- 

 tions of motion will still be that described in p. 207. These equations do not, 

 as we have seen, admit of an integral representing plane wave motion. It is 

 easily shown that the difficulty here alluded to does not affect that part of the 

 theory of light or sound in which the direction of the reflected or refracted ray 

 is derived from the consideration of wave motion. 



16. Before concluding the present Memoir, I think it necessary to say a few 

 words on the applicability of the integral calculus to problems like the present, 

 or more generally to any problems in which bodies are considered, not as con- 

 tinuous masses, but as assemblages of distinct molecules. 



I may remark, in the first place, that the method and results of the present 

 Paper would be in no wise affected by the rejection of the molecular hypothesis; 

 all that is essential to the vaUdity of the method here given being attained 

 by defining a molecule to be a particle so small, that the motion of the system may 

 be fully represented by the motions of all these particles considered as units; and 

 without such a supposition no equations of motion of a continuous body appear 

 to have a perfectly definite meaning. 



But as the constitution of the bodies which we find in nature appears to fa- 

 vour the supposition of separate molecules, rather than that of perfect conti- 

 nuity, it becomes an important question to determine how far the methods of 

 the integral calculus are applicable to such cases. This is the more necessary, 

 as M. PoissoN denies the appHcabihty of these methods to any problems con- 

 nected with molecular force; and, more generally, to any problems in which the 

 force varies with extreme rapidity within the Umits of integration : — " Au reste 

 la formule d'EuLER qui sert h transformer les sommes en integrales, contient une 

 serie ordonnee suivant les puissances de la difierence finie de la variable, qui 

 n'est pas toujours convergente, quoique cette difference soit supposee tres petite. 

 L'exception a lieu surtout dans le cas des fonctions comme /(r) qui varient 

 tres rapidement."* 



It is quite true, that the methods of the integral calculus are in strictness 

 applicable only to continuous masses, and that it is in such cases only that the 



* Mem. de I'Inst. torn. viii. p. 399. 



