FREED FROM THE ACTION OF GRAVITY, 31 
figure will constitute the final state in question: for it may 
happen that several figures of equilibrium corresponding to the 
same primitive disturbance might equally possess stability, and 
that the choice of the mass for one of these figures may have 
been determined by other circumstances; for example, by the 
modifications which its movement experiences in the first mo- 
ments of rotation. In fact, it is by examining these modifica- 
tions, to which the attention of geometricians has not been di- 
rected, that I shall attempt to arrive at the mode of generation 
of annular figures. 
16. When the mass begins to revolve upon itself, the angular 
velocity of the portions remote from the axis, which are carried 
off by their centrifugal force, necessarily goes on diminishing. 
This diminution is especially apparent on the equator of the 
mass, and it is the more considerable in proportion as the ini- 
tial movement of rotation was more rapid. It thence results 
that in the first instants of a sufficiently rapid rotation there will 
be a great difference of angular velocity between the portions 
which are near the axis and those which are near the equator. 
Nevertheless, if we admit for a moment, that in virtue of the 
adherence of the liquid for itself, and of the friction of its several 
parts, the portions which turn most rapidly communicate by 
degrees a part of their velocity to the others, so that in the end 
the result is a mean angular velocity corresponding to the same 
moment of rotation, and equal in all the points of the mass, this 
may take an ellipsoidal figure. But long before the feeble forces 
of which we have just spoken can bring about this mean result, 
another order of phenomena would be manifested, which may 
impede the development of the elliptical figure and give rise to 
an annular form. 
In fact, it follows necessarily from the preceding considera- 
tions that, in the first instants of a rotation sufficiently rapid, 
the centrifugal force at the equator of the mass will be much 
less than that which would correspond to the above mean velo- 
city ; and that, on the other hand, the centrifugal force of the 
portions near the axis will be by much superior to that which 
would correspond to the same mean velocity. The liquid next 
the axis will therefore be driven towards the liquid of the equa- 
tor, whence there will necessarily result the formation of a sort 
of circular cushion (bowrrelet), more or less marked. In other 
words, the mass will soon become hollow in the middle, and will 
