122 The Rev. S. Haughton on a Classification of Elastic Media, 



contain all that is requisite to determine the motion of pomts situated in the 

 interior of the mass. The mechanical conditions to be satisfied at the limits 

 are contained in the double integrals, which become, by the substitution of the 

 element of the bounding surface, 



]\pdS {Ix cos X + Sy cos /L£ + 8^ cos v), 



(\, n, v) denoting the direction of the normal. It may be easily shown from 

 this expression, that if P denote a function of {a; y, z, t), expressing the normal 

 forces applied at each point of the surface, the mechanical condition at the limits 

 will be expressed by the equation, 



p-P = 0. (23, b) 



To which must be added the equation of the bounding surface, deduced from 

 geometrical considerations, 



dF dF dF dF ^ /o. ,x 



dt dx dy dz 



It is unnecessary to enter further into this subject, as it is fully treated by 

 Lagrange, in the second volume of the Mecanique Analytique, and is indeed 

 the only example given by him of the application of his formulas to the problem 

 of bodies composed of continuous points. 



Resuming the former signification of (x, y, z, ^, »;, f ), we find from equations 



dV 

 (5) or (15), since 2 F = 2pw + Aw-, and -j- = p + Aw, 



d-$ _dp . dai 

 dt^ ~ dx ^ dx ' 



If no external forces act, p and A will be constants ; and, in this case, these 

 equations become, assuming a negative sign for A, so that the equilibrium may 

 be stable, 



