and the Laws of Plane Waves propagated through them. 117 



(dV dV ^ \ ^ . ^ 



V^' ^' ^°7 ^''^noting forces tending to alter the quantities (a„a„a3, &c.), 



which are functions of {x, y, z). The whole body may, therefore, be divided 

 into couches by a series of surfaces wliose equation will be 



C denoting the parameter of the system. In a similar manner, the body may be 

 conceived as divided into couches by other sets of surfaces corresponding to 

 («2, "3, &c.). In any one of these couches the corresponding func°ion 

 (a,, a,, &c.) will have a constant value, which will vary from one couche to 

 another. 



If i = be the equation of a surface, we may conceive all space as divided 

 into two portions, which will be distinguished by the property, that the portion 

 lying at one side of the surface will have the function of {x, y, z) denoted by 

 (Z), positive ; while for the rest of space, lying at the other side of the surface, 

 the function {L) will be negative. Similarly, cL will be positive for all dis- 

 placements made at one side of the surface; negative for all displacements on 

 the opposite side ; and zero for displacements in the surface itself. 



Let us now resume the equation, a, - 6'= 0, which denotes a surface drawn 



in the interior of the body, along which the value of |i remains constant. It is 



necessary for the stable equilibrium of the body, that if the particles composing 

 this surface be displaced from it, the molecular forces developed by the dis- 

 placement should tend to restore them to the surface ; this condition will 



. ,1 ^ (dV dV ^ \ , . , 

 require tiiat ^_,_,&c.j, which are the forces developed by the displace- 

 ments (ga,, ia^, he), should have signs opposite to those of the displacements. 

 If no forces are developed tending to restore the molecules, we shall have 



T- = 0, ^- = 0, &c. : 

 ««! da, ' 



and these equations will determine the /iWfe of stable and unstable equilibrium 

 in the body. 



