CHAPTER VIII 
THE CONFIGURATIONS OF ROTATING LIQUID MASSES 
which are subjected to forces such as to make the spherical shape impossible. 
are acted on by no forces but their own gravitation, confining ourselves 
to the simplest case in which the matter is supposed to be homogeneous and 
incompressible; the far more difficult problem presented by masses which are 
compressible, and so are not of uniform density, is reserved for the next 
chapter. Even the simplified problem has taxed the skill of some of the most 
eminent mathematicians. Among those who have specially worked at it may 
be mentioned Maclaurin, Jacobi, Lord Kelvin, Poincar6, Sir George Darwin 
and Liapounoff. 
188. If a homogeneous mass of matter of uniform density p is rotating 
about the axis of z, the conditions that it shall rotate in relative equilibrium 
as a rigid body rotating with angular velocity w are expressed by the system 
Here p is the pressure at the point x, y, z, while V is the gravitational 
potential. The equations are of course merely the ordinary hydrostatic 
equations we have used before, modified by the additional terms co 2 px, (o 2 py, 
which represent the effect of centrifugal force. It is not necessary to specify 
the mechanism by which the pressure p is exerted. It is the total pressure, 
including pressure of radiation if this is appreciable, and there is no need to 
pecify it further. 
Since p is supposed to be constant throughout the mass, the three equations 
have the common integral 
The pressure p, however it arises, must be zero over any free surface of an 
astronomical body. If this condition is satisfied, equation (188-4) fixes the 
187. We proceed to apply the general principles just explained to the 
problem of determining the shapes assumed by astronomical or other masses 
In the present chapter we deal with the shapes of rotating bodies which 
General Equations. 
of equations 
(1881), 
(188-2), 
dp _ d V 
dz P dz 
(188-3). 
P 
(O 2 ( X 2 + y 2 ) + cons. 
..(188-4).