AND THE FIGURE OF A HOMOGENEOUS PLANET. 
127 
As the centre of gravity of a mass of fluid in equilibrium must be free from 
the action of any force, except what is common to all the particles ; and as the 
attractions of the particles balance one another at that point ; the centrifugal 
force must likewise be evanescent at the same point, and consequently the 
axis of rotation must pass through it. Let three planes intersecting at right 
angles, one being perpendicular to the axis of rotation, pass through the centre 
of gravity; and assuming any particle of the fluid, let r denote its distance 
from the same centre, and x, y, z its coordinates, z being parallel to the axis of 
rotation : further, let g represent the attractive force of the whole mass of fluid 
at the distance equal to unit from the centre of gravity ; and ,/the centrifugal 
force (that is, its proportion to g) at the distance equal to unit from the axis 
of rotation : then g r will be the central attraction urging the particle, and g x, 
gy, g z, will be the resolved parts of the same force in the directions of the 
coordinates ; also, 'x 2 + y 2 will be the distance of the particle from the axis 
of rotation ; — f x 2 + y 2 , the whole centrifugal force estimated as tending 
to shorten the coordinates ; and — fx, — fy, the resolved parts of the same 
force, parallel to x and y : collecting, now, the partial forces which urge the 
particle in the respective directions of the coordinates, we shall find, 
x =(g ~f) ^ Y = (g —f)y> z = gZ‘ 
The equations of equilibrium will, therefore, be 
<p =f{Xdx + Ydy + Zdz) = $ {(g — /) (x 2 + y 2 ) + gz 2 }, 
V = C - •§ { (g -f) (x 2 + y 2 ) + g z 2 } 
The equation of the surface of the fluid will be found by makings = 0, viz. 
c = i (g ~ f) 0 2 +y 2 ) + %gz 2 - 
_/ 
a 2 — x 2 + 3/2 + - 
And, if we put e 2 = — , the same equation may be thus written, 
o 
which belongs to an elliptical spheroid of revolution having the equatorial 
semidiameter equal to a, and the polar semi-axis to a ^/l — e 2 . 
8 . The order of discussion that has been laid down now brings us to the 
