172 Compressible and N on- Homogeneous Masses [OH. vn 



Thus if we assign to 6 the value given by equation (454), it is clear that 

 our general equation (450) will reduce, when the mass is incompressible, to 

 the equation from which the solution for the incompressible problem was 

 previously obtained. With this meaning for 0, the general equation (450) 

 becomes 



*] + 2^ 



The solution for the incompressible mass is derived from the equation 



- [2 $] -- [J^ + J*f + Jc*\ + ^ c <* + y V -(458), 

 which is a special case of the above. 



172. On equating coefficients in this last equation we shall obtain 

 three equations (456) and the solution of these equations will consist of sets 

 of values of a, b, c and to 2 . 



Similarly the solution of equation (457) will consist of sets of values of 

 a, b, c, ft> 2 and P . In the incompressible problem, P is always zero, and the 

 sets of values for a, b, c and o> 2 coincide with those found from equation 

 (458). But in the more general problem, this is not the case. 



Let us now agree, as a matter of convenience, that the symbols a, b, c 

 shall be reserved to refer only to solutions of the incompressible problem. 

 A solution of the compressible problem may now be designated by symbols 

 such as a + Aa, b 4- A6, c + Ac. Strictly speaking, the equation of the 

 boundary ought no longer to be taken to be 



2- 2 + eP = l ........................... (459); 



a 



it must be taken to be 



(a -I- Aa) 2 



This however may be re-written in the form (459) if we permit of P 

 containing terms of degree 2 as well as those of degrees 3, 4, ... of which it 

 has so far been supposed to consist. Thus, in what follows, we shall 

 suppose P to include second degree terms and a, b, c will be supposed to 

 have the meaning just agreed upon. 



Let the general value of F;(l) corresponding to the boundary (459) in 

 which P consists of terms of degrees 2, 3, 4 ... be supposed to be 



Vi (1) = - Trabc (J A x* + J B f + Jcz* -J) + 



