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



d-ij , d-^ . 

 dt- ^ dx- 



d't , d'^ „ 

 dt^ ' dx' 



Of these equations the second and third are deduced from the first by 

 simply changing, in the suffixed letters, a into /3' and 7' respectively. 



If the body be homogeneous, i. e. if all its points be absolutely similar, the 

 quantities 



A^%^ , &c., B^-.^ , &c., i\^„., &c., 



will be evidently constant. The terms involving 



d^ n d)] „ d^ „ 



^''^"■' S'^°-' ^'^'^•' 



will, therefore, disappear, and the equations (N) will become homogeneous 

 partial differential equations of the second order with constant coefficients.* 

 The number of constants which these equations contain is the same as the 

 number of terms in the right hand members, namely, eighteen for each equa- 

 tion. Hence it is evident, that the equations which represent the small oscil- 

 lations of a homogeneous medium satisfying the hypothesis of independent 

 action, contain, in general, fifty-four constants. This is the greatest number of 

 constants which these equations could be made to have without a change of form. 

 6. The conditions to be fidfilled at the limits of integration are, of course, 

 obtained from the terms which appear under a double sign of integration in 

 the equation derived from (I). These terms will evidently give 



\\{Pf^-\-Q,i^} + R,K)dydz 

 + [( (Poc^ + Q.,?./ + E.l^) dzdx (NO 



-1- IJ (P3CI + Q^c>j -t- Ri^) dxdy = . 



Let p, q, r be the angles which the normal to the surface bounding the 

 given medium makes with the axes, and let dS' be the element of this surface. 

 Then, if equation (N') be transformed in the usual way by making 



* This conclusion does not hold for molecules situated at the surface of the body. — Vid. Art. 15. 



