140 Compressible and Non- Homogeneous Masses [OH. vn 



these equations of equilibrium become 



dp dl 



= p ............ (392), 



dy r dy 



S / = p~ ...(393). 



dz r 02 



On equating two values of d*p/dyd* we obtain 



dp d&_dp 9ft 

 dz dy dy dz ' 

 so that 



dp dp dp 



doc dy dz 



........................... (394) ' 



dx dy dz 



It follows at once that the surfaces p = cons, necessarily coincide with the 

 equipotentials H = cons., and it further follows from equations (391) (393) 

 that these surfaces also coincide with the surfaces of constant pressure 

 p = cons. The boundary of the fluid must of course be one of this family of 

 surfaces, say p = a, and the necessity for the condition that H shall be 

 constant over the boundary, which has so far been used as the condition for 

 equilibrium, is at once obvious. 



The condition that O shall be constant over the boundary will however 

 no longer be sufficient to ensure equilibrium ; it is still necessary but not 

 sufficient, and equations of equilibrium must be satisfied throughout the 

 mass. 



Masses of Uniform Composition 



140. The simplest case arises when the matter is of uniform composition 

 throughout, so that the pressure is a function of the density, say 



?-/</*) 



Equations (391) etc. now assume the form 



df(p) dp an 



o 1 ^ f = P o etc - 



dp ex r ox 

 If <f> (p) is defined by 



p dp 

 these become 



(395), 



_ ,~ 



dx ox 



