557 
figure of the earth.- 
( 1 1 ) . Collecting the terms now in the same manner as in 
(9), according to the directions in (6), and observing the 
values of the integrals in (7), we find that part of V which de- 
r d. a *(i + i£j_lL_ -im- 
pends on the exterior strata = — / P 1 Li — 
r W / da 
T r :f 
d(e 
-J-e’ + i.) 
H 7 / 
d a 
2 1 .4 v . C a , 4 0 . 2 i 3 
d.a*(i + ±l + 
3 15 15 
d a 
Let 
/ 
d (e — Le 4 + ^ ) 
% ( # ) ; /'p*y--4-=<r(tf); t ^ ien taking the integrals from 
J a aa ' 
a = a to a = a we have for the exterior strata, 
V = ^U {r(a) — 
(12). Adding together the expressions in (9) and (11,) we 
find for the complete value of V, ~ f | {r(a) — t(«)} -{- 
(^ + r°{x(a)-xM}) . ^ + (^ + 7^ {»(a)-r («)}) . 
[fo — 7 ^ + T •‘I } 
(13.) When a fluid is in equilibrium,* if x,y, and z, be 
the co-ordinates of a point in it, and P, Q, R, the forces in 
those directions, then at a surface of equal density P £ x -f- 
Q£y4"R^ %=:::0 (liv. i. n°. 17) ; and the equilibrium is 
impossible except Ptla? 4* -f- R < 5 “% be the complete 
variation of some function U. The equation then to a 
surface of equal density is U = C. In the present instance, 
* I have not considered the second condition of equilibrium, given by Mr. Ivory 
in the Philosophical Transactions for 1824, as the reasoning upon which that Gen- 
tleman has founded the necessity of such a condition, appears to me altogether 
defective. 
4 C 
MDCCCXXVI. 
