In this he showed that a homogeneous fluid mass revolving uniformly round an axis under the action of gravity ought to assume the form of an **ellipsoid** of revolution.

The cases of greatest practical importance are those of a sphere (which is an **ellipsoid** with three equal axes) and an ovoid or prolate **ellipsoid** of revolution.

When the **ellipsoid** is so much elongated that I is negligible in relation to m'-, the expression approximates to the simpler form N=412 (log 201-I).

Laplace treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an **ellipsoid** of revolution whose equator was determined by the primitive plane of maximum areas.

As an application of moving axes, consider the motion of liquid filling the **ellipsoid**al case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the **ellipsoid** l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the **ellipsoid**al case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.