220 FIGURE OF THE EARTH. [11 



densities p lt p z , ... p n are known, and also their volumes, and let them rest 

 upon a solid nucleus of known ellipticity e . Let M be the mass of the 



nucleus, -TTK/ its volume, and let E O = -TT \ p^^dK. Further, take 



- 



Then exactly as before we have the equations 



xf i _^o_4 7j ,, -|fS sj_ ,4^ / e _ 6 \ K! _ 



K! /q 3 5 K/ 5 



5 "" ^ 2 



^yW Af lf^_^?_l 5 5 e \ -TT (/c 5 e K M 



_ 4 , ., _ 1 2 2 _ 



6, -E^o 4 1 4 1 



^"1 ^H 3 ^"i 



.!. ( K _ K )l_i,r ( e _ ) K '_ _i ff ( _ ) K 2 -A ft) K a = 



5 Ko* 5 5 



n _ - 



K n-l K -i 



4 11 



T ^Pn (^n - ^-1^-1) , ~ o <>?Kn = 0. 

 /C n L 



By means of these equations the n quantities Cj, e 2 , ... e M may be deter- 

 mined. The process of solution may be simplified if we divide in succession 



by /, K/, ... K n 2 respectively, and write for brevity M^-\- M^+ ...+M i = S i . 

 Then we shall have 



---- - 00 + ^^ - --i = 0, 



f 2 K l 



S 3 S, 7 1 1 \ 4 . / 1 IN 



3 



/C 2 



I 14 /l 1 



5 - - = 0, 



/C 3 



