AND THE FIGURE OF THE EARTH. 
91 
16. To find the equation of equilibrium of a heterogeneous spheroidal mass of fluid, 
revolving about its axis, with an angular velocity a. 
By the principles of hydrostatics the general equation is = J * (X dx + Y dy 
+ Z d z), §' being the density at the point ( x , y, z), p the pressure, and X, Y, Z the 
d V d~V d Y 
sums of the resolved parts of the forces ; which are — and the 
centrifugal force. Let the axis of z be that of rotation ; then the centrifugal force is 
u 1 x along x, and co 1 y along y. Let us express u in terms of the ratio of the centri- 
fugal force at the equator to the equatorial gravity. Call this ratio m, which is small 
in the case of the earth, being of the same order as z. Then 
0 4 7r q m ip a 
- — 
m 
mass 4 n % 4/ a 
, or a* 
Therefore 
and 
d Y 4 7r gm\J/ a .x v dV 4 t: a .y „ dV 
dx ' a 3 , dy ' a 3 ’ dz 5 
fij, = v + 2^i2 (1 _^ )r2 . 
Now n is a constant for a level surface. Hence for any stratum we have 
c = V +^‘(i- P V. 
At the surface this is 
C = *r ~ 3 -rS‘ <t> d (“' 3 0 - f 0 - * a ' d ^^ + ?£(•- f" 2 > 
= ,>«- ~ t) Na + IfS Ma (* - 
where 
a/t r a ± ( a ' 3 (! - o) i , , xt r a ^ i d ( a ' 5 s ') a / 
M a = y (pa 1 — — — - cl a, and N a = J ^ (pa 1 da , d a. 
For r write a (1 — s (m 2 ), then 
Mg „ 1 / 2 l \ N a 1 2 \ M a 
c = Ti 0 + s ) ~ T V‘ i “ -s)yF + T m 0 - f ‘4 -r • 
Equate the coefficients of y> 2 , then 
Ms/ m\ 3 N a /T ,. 
17- By differentiating and changing the sign of we obtain the amount of 
gravity which acts towards the centre ; which, to the order we are now considering, 
is the same as the whole force of gravity ; since the cosine of the angle of the verti- 
cal differs from unity only by terms of a higher order. 
n 2 
