11] FIGURE OF THE EARTH. 219 



and the term I w V - 1 ffl v (x* - i 



o 2 \ o 



must be added. We obtain the condition 

 (III) (If, + M,+ M 3 ) 1< - ^v, 1 - vp, (^ 



The three equations (I), (II), (III) serve to determine the quantities 

 e 15 e,, 6,. Write 



and divide the equations by K*, K.?, K S " respectively. Then 



$4 4 4 ,1 



-; e, - - wyv, - - 7rp 2 (e, - ej - - n/), (e 3 - e 2 ) - - at = 0, 



& 4 < 4 * \ 4 



, , 

 , B t 3 (e 3 -e 2 )--a) = 0, 



4 Kl ' 4 //c 2 5 K, 6 \ 4 



2 ,-,6, -^ 



K.j J \K 3 K 3 / y /C 3 



Take the differences (II) -(I), (III) -(II): 

 , 5, 4 . / 1 1 \ 4 /I 1 \ 



C2 ~ K 3 l ~ 5 7r/)ll ' Cl ' 7 5 ~ K V + 5 7r/>26l ' C 



K l *J \ f 3 /Cj / J 



, 



4 . 1 1 \ 4 . / 1 1 



i ^S 3 C 2~ C "^l^l"-! 



K-e, t/ 



4 . /I 1\ . 4 



5 

 or, as they may be written, 



S a S, 4 . / , / 1 1 



+ - ir^ic, 8 -, - -i + 



From the former we find e 2 in terms of 6j and known quantities ; sub- 

 stitute in the latter and e 3 is found in terms of e lf and e t is then found 

 from (I). The case when there is a solid nucleus may be dealt with in 

 the same way. If e be the ellipticity of its outer surface, e is known 

 and there is no condition of uniformity of pressure at its surface. Pro- 

 ceeding with the series, 1 may be found in terms of or and e , e 2 in terms 

 of e,, e , w 2 , and so on. 



13. Now take the more general case of any number of fluids whose 



282 



