11] FIGURE OF THE EARTH. 223 



.. , d*e 2 dMde dM e 6e 



it becomes -?-; + if> -/ r + 2 -v mr r 0, 



1 M dK dK dx. MK /c 2 



K die 



Or ;rv -j = j J 



M dK ae e 



CtK K 



the right-hand member is supposed known in terms of K. Integrating we get 



6e d*e 



2 T^ 

 K' UK' , 



-? dK; 



d L+* 



CtK K 



this will give M, and p then follows from the equation 



1 dM 

 P ^ 



47TK 2 Ct/C 



15. Let us assume a particular law of density and find the associated 

 law of ellipticity ; let us take 



. sin qK 

 p A - ; 



K 



at the centre where * = 0, this gives p = Aq, and a gradual diminution as we 

 ascend. Then 



dp A , 



-y- = (qK cos qK sin qK) ; 



UK K' 



dM 



and - 7 = 4,TrpK' 4:irAKSin qK, 



UK 



M= - : {sin OK OK cos OK], 

 1 



47TK 2 rfp 



and -ji, ~r = ~ < 



M. CIK 



Therefore the equation becomes 



a case of Riccati's equation of which the solution is 



A 



Me = -,-5 {(3 - Y) sin (qK + a) - 3qK cos (qK + a)}. 

 qK' 



The two arbitrary constants must be adapted to the values of E^ and of. 



