11] FIGURE OF THE EARTH. 221 



S n . S n _ , i 1 1 A 4 _ . / 1 1 



v 3 n-1 \K- 5 if s I K "i \ i i ' Of \ 5 5 



" n-i \ K n K n-i/ \ K n K n-i/ 



4 



= 0. 



K -! n-in-in-i-, 



The first of these gives e 2 in terms of e, and e , the second gives e 3 in 

 terms of e,, e lt e fl , and so on; by this means we may express all in terms 

 of e,, which may then be determined from the equation 



S l E, 4 5 . 1 4 , 4 1 



^3 e i~ ~ 5 wpi ( K I i~ K o o) 5-5 ^P 2 (.-i) ----- 5 "Pn ( e - e -i) ~ g w2 = - 



14. Let us now pass to the case where each layer of fluid outside 

 the nucleus is indefinitely thin. We shall then pass from sums to integrals 

 and from difference equations to differential equations. 



Let K be taken as independent variable, and let p be known in terms 

 of K ; then if M be the mass interior to the stratum defined by K, 



M=M + 



I 



J * 



and it is required to determine e in terms of /c. Then we have at any point 



v- **_ E. A 2 _ A _ 4 TT / _ i\ f d 



~r r s \ 3 5 " 



( K i 4 - A. l \ ( K d * , 



47T PKCIK--TTT A. 2 -- p -j- da, 

 J* 5 \ 3/ J K r rf/c 



where K is the value K assumes at the external surface ; the second term 

 on the right being obtained from the nucleus, the third from the mass 

 internal to the point, and the fourth and fifth from the mass external to 

 the point. Substitute for l/r its value 



K, 



I \ /J / 



and we get 



K 



477 



\ 



i\ fK 



-^ 



O/ J 



where (7 is a quantity independent of X, but varying with K. Hence we 

 have the two equations 



M ( K 1 



- + 47T O/CCZ/C + - ft)V = C, 

 K J K 3 



, e " 4 7T f' d , .,j 4 A K <-U -. \ 



M --- jT--J P-T (t)d K --TrK- I p -, rf/C-- 

 /C K 3 5 K 3 CZ/C V 5 J, K 



