GRAVITATIONAL STABILITY OF THE EARTH. 219 



hydrostatic pressure. We suppose also that the law of elasticity of the body is that 

 the increment of pressure is proportional to the increment of density. We show 

 that equilibrium is possible in strained states, in which the excess of density at any 

 assigned distance from the centre is proportional to a spherical surface harmonic of 

 the first degree. 



In the initial state the pressure p a and potential V are given by the formulae 



Po = |T>Y>.' (a'-S), V = firypo (3a'-r). 



In the strained state the pressure P, density p, and potential V are expressed by 

 the formulae 



where denotes the condensation. The equations of equilibrium are 



av ap av ap av ap 



'-" 0) p - = ' ?--'- 



and W is connected with by the equation 



When terms of the second order in the small quantity f are neglected, the 

 equations of equilibrium become three equations of the type 



-x-Oi ....... < 108 > 



and, on eliminating W, and writing s* for $7ry/> 2 /X, we have 



V'+sV | + 6^=0 ......... (109) 



or 



This equation is satisfied by putting 



where A is an arbitrary constant and <u, is a spherical solid harmonic of the first 

 degree, and this is the most general form of solution in which f is finite at r = 0, and 

 is proportional when r = const, to a surface harmonic of the first degree. The 

 additional potential W has the form 



W = ^^{lAa-'e-i-X+F,}, 



where FI denotes a spherical solid harmonic of the first degree. 

 Let the bounding surface become 



r = a + U.. 

 2 F 2 



