HOMOGENEOUS FLUID AT LIBERTY. 513 



equilibrium must coincide. The diameter parallel to .r' being thus proved to be an 

 axis of the ellipsoid, we may assume that the other two coordinates are parallel to the 

 remaining axes of the geometrical figure, in consequence of which the equation of 

 the surface will become more simple, viz. 



the three semiaxes being k, k\ k\ of which k is the axis of rotation. 



Further, the figure of the fluid in equilibrium can be no other than a spheroid of 

 revolution. Draw a plane through the axis of rotation and any point {x y %) in the 

 surface of the fluid. This plane will contain that part of the attraction of the spheroid 

 which is parallel to the axis of rotation, or to the coordinate x : it will also contain 

 the centrifugal force directed at right angles from the axis of rotation. The same 

 plane will also contain the resultant of the attractions parallel to ?/ and z ; for if it did 

 not, the resultant might be resolved into two forces, one contained in the plane, and 

 the other perpendicular to it ; and the force perpendicular to the plane would partly 

 act iii a direction touching the surface of the spheroid, which is inconsistent with the 

 equilibrium of the fluid. Wherefore, the whole attractive force at any point in the 

 surface of the spheroid is contained in a plane passing through the point and the axis 

 of rotation ; which obviously excludes ellipsoids with three unequal axes, and limits 

 the figures of equilibrium to spheroids formed by the revolution of an ellipsis about 

 the axis of rotation ; and as the centrifugal force necessarily causes the equatorial 

 diameter to be longer than the polar axis, it follows that the figure of the fluid in 

 equilibrium can be no other than an oblate elliptical spheroid of revolution, of which 

 the equation is 



^;2 

 ^^ = cT^ + ^ (3/2 + ^2), 



the fluid turning about k, the less axis. 



By the foregoing investigation, the problem for determining the equilibrium of a 

 homogeneous planet in a fluid state is reduced to solving the equation of the upper 

 surface, which is an expression of a known form, as the figure of the fluid is ascer- 

 tained. The equation of the upper surface adjusts the oblateness of the spheroid to 

 the quantity of the centrifugal force. It is only this part of the problem which, if we 

 judge rightly, is fairly made out in the modes of investigation usually adopted ; for in, 

 all these it is assumed that the figure of the fluid is an oblate elliptical spheroid, but, 

 except in Maclaurin's demonstration, the equilibrium is not proved on satisfactory 

 grounds. D'Alembert first observed, that in general more spheroids than one may 

 be in equilibrium with the same centrifugal force, or with the same velocity of rota- 

 tion ; and it is now well known that there may be two such spheroids, or one only, 

 or that no spheroid of the proposed matter can be found that will be in equilibrium 

 with the given quantity of centrifugal force. All this is pure mathematical deduction 



