figure of the earth. S59 
Now if e and E are the same functions of a which e and 
A are of a, ~ to the second order = -- ( i — 6 • * ”(^ 2 4* 
i . i — — — y -{- E . j* 2 — [x 1 ) : ^ to the first order = ^ 
( l — 3 e • i — f* 2 ) ; r 2 = a 3 (l -f- 2 £ . l — (x a ) ; ~ including 
no small quantities = ~ ; r*= These are not to be taken 
farther, because ^ («), %(a), and are of the first order, 
and u(«) and <r(a) of the second order. Substituting these 
values in the last equation, we find C = 
■3^ 
3 * 
2 T 2 
cr . 1 — |X 2 
^ + K r W- T «} 
+ {*(«)-*(«)}). 
+ ^ .7=? . , .. rr 
+ ».rf {*(»)-*(.)} i^^+^+j^W-rW}) . (a_£ k . + 
Making the coefficient of (x 2 = o, and that of o, and ob- 
serving that, as the equations which we shall find are general 
for all values of a, we may put a for a, 
f- • *-£—%* { xto-xtoy—fa* 
+ "7 7 a ‘ { <r ( a )~ <r (‘ J )}) 
These are the equations of equilibrium ; and since by dif- 
ferentiation they may be reduced to two differential equa- 
tions, from which the two quantities A and e are to be found. 
o = 
