GRAVITATIONAL STABILITY OF THE EARTH. -J23 



then equation (112) becomes 



+y+ C. . (us, 



The left-hand member of this equation is the same as that of equation (107) in 47 

 above ; and therefore, when to* is neglected, f can be of the form &, where 



the notation being the same as in 48. Now we shall suppose that o>* is not large, 

 so that we may treat , as an approximation to , and substitute & for f in the 

 right-hand member of equation (113), for all the terms of this member are small of 

 the order * We are then neglecting f*, but not w* To obtain a second 

 approximation, we put 



f- 



and seek a particular integral of the equation 



There would be no special difficulty in obtaining a solution of the equation, but it 

 will be sufficient for our purpose to find the form of the solution. The function , 

 may be expressed in terms of polar co-ordinates r, 0, <f> in the form 



f, f(r) r (a, sin cos <f> + /3i sin sin < + yi cos 0), 



where a,, /?,, y, are constants, and ./'('') is a certain function of r which has been 

 determined. Hence we have 



r/(r)(, sin cos <j>+ 0, sin sin <) 

 ox cy 



+ i*f (r) sin 2 (i sin ^ cos <j> + /8, sin ^ sin <f> + y l cos 0), 



z > = r /(r) y, cos ^+ r/' (r) cos* (a, sin cos ^+0, sin sin < + yi cos 0) ; 



oz 



and these can be expressed in the forms 



8n +'' r 



-/-/' (r)(cos s 0-|) sin 0(a, cos <+/8, sin ^) -ry'(r) y t (cos 3 0- f cos 0), 



+ r*/'(r) sin 0(cos 3 0~i)(, cos <+& sin <^) +ry(r) yi (cos 3 6-$ cos 



Hence the right-hand member of (114) can be expressed as a sum of terms each of 

 which is the product of a function of r and a spherical surface harmonic, and the 



