76 The Gravitational Potential of a Distorted Ellipsoid [CH. iv 



Thus collecting results, we find for the second order terms, 



. + A = Q - * DP * +f\ 



Proceeding in the same way, we find for the third order terms*, 



Us +/^3 = U 3 +f(w s 



~ t # 2 (PQ) + 



and this completes the solution as far as the third order of small quantities. 



76. The solution which has been obtained is found, on collecting terms 

 to be 



$ = e(u l +/^) + e> (u. 2 +fv 2 ) + e 3 (u 3 +fu s ) 



+ # [Q - 



+ < [R - IDPQ + Tfa&P* - \f f 



Putting X = 0, the value of < at the boundary is seen to be 



< = eP +^ + e 3 ........................ (198), 



and since P , Q , J^ are entirely at our disposal, this value of c/> is capable of 

 representing a general distortion of the fundamental ellipsoid as far as the 

 third order of small quantities. 



This same distortion might of course have been supposed to be merely 



<o = *Po ................................. (199), 



it being at once possible to pass to the general form (198) by replacing P 

 by P + eQ Q + e 2 R . We have introduced Q and -R separately on account of 

 the limitation which has been imposed that P shall not be of degree above 

 the third. 



77. When. this limitation is removed, equation (199) may be regarded 

 as representing the most general distortion of any kind which can be ex- 

 perienced by the fundamental ellipsoid. By analogy with equation (197) 

 the corresponding value of < is seen to be 



e P - \fDP + 



- 4<c [DP* - 



+ T J 5 ' [&P> - 4/Z> 3 P 3 + d^/'-D'P 3 - ... J etc ........... : .(200). 



* For details of the calculation, see Phil. Trans. 217 A (1916), p. 7. 



