GRAVITATIONAL STABILITY OF THE EARTH. 



175 



the particle which was initially at (x, y, z) is displaced to (z+ti, y+v, z+w), the stress 

 is specified by six stress-components X,, Y y , Z., Y,, Z z , X,, and these are connected 

 with the initial pressure /) and the displacement (u, v, w) by the formulae 



3t* 



ov 



. . (1) 



where X and /* are constants. It is required to form the equations of vibration, and 

 to solve them, so as to determine the character of the modes of vibration and the 

 equation for the frequencies, and, in particular, to ascertain the relations which must 

 hold among the quantities X, p., p^, a in order that any frequency may be reduced to 

 zero. We proceed to express this problem in terms of a system of differential 

 equations which hold at all points of the body, and a system of special conditions 

 which hold at all points of the undisturbed surface. 



3. In the equilibrium state the potential V at any point is given by the equation 



-r), ......... (2) 



where y is the constant of gravitation. The equation of equilibrium is 



_!3o + 3V, ...... (3) 



po dr 3r 

 or 



|r=-^*V. .......... (4) 



Since p* = at the surface r = a, the value of p* at any point is given by the 

 equation 



^lirypo'Ca'-r*) .......... (5) 



When the sphere vibrates, the equations of motion are three equations of the type 



3 s ?* 3V . 3X, _,_ 3X, ^ 3Z, 

 />3rf = P -5- + -5-* + -*-*+ -3-*, ....... ( 6 ) 



3r r dx ox ay cz 



where p is the density, and V the potential, in the disturbed state. In the left-hand 

 members of these equations we may ignore the distinction between p and fv In the 

 right-hand members we may put 



p^oO-A) ....... .-. . ... (7) 



