and the Laws of Plane Waves propagated through them. 107 



^„ ,, dI\dl\dP, 



dQ^ dQi dQ, 



, „ , . dRi dRi , c?/?3 



These are identical with equations (2), and are true for all kinds of mole- 

 cular action. In the particular case of a rigid parallelepiped, we should intro- 

 duce another set of conditions arising from the equilibrium of couples. Let 

 (Ldm, Mdm, Ndm) be external couples applied to the parallelepiped; these 

 must equilibrate the couples arising fi'om the molecular action of the sur- 

 rounding parts of the body ; it is easy to see that, neglecting the small forces 

 arising from the differential coefficients of P,, Pi, &c., the couples round the 

 axes of X, y, z, will be 



(i?2 + Qs) dydz, {P, + R,) dxdz, ( Q. -t- P^) dxdy. 



Hence the required conditions will be 



eL^R,+ Q, ; 

 eM=P, + R,; 



and, if no external couples be applied, the conditions will be, 



R2 = Qs] 



P, = R, ; (9) 



since the couples (R.2, Q^), &c , must act in opposite directions. 



These conditions (9) were given by M. Cauchy,* and afterwards adopted 

 by M. PoissoN.f These writers seem to have considered them as necessary for 

 all systems ; but this is not true, as equations (8) exhibit all the relations which 

 exist between the forces and the motions produced. Equations (9) are necessary 



* Exercices de Mathematiques, torn. ii. p. 47. 



t Journal de I'EcoIe Polytechnique, cahier xx. p. 84. 



p2 



