142 Mr. Ivory on the figure requisite to maintain the equilibrium 
we seek the value of xthat will make q a maximum, we shall 
find this equation, viz. 
Arc. Tan. x = 
* 9 + 7* a . 
I + & ' 9 + ’ 
from which x comes out equal to 2*5292. And hence Vi + x a , 
which is the proportion of the equatorial diameter to the 
polar axis, is equal to 2.7197. 
From all this it follows, that if a homogeneous mass of fluid 
in equilibrio, at rest, and consequently of a spherical figure, 
begin to revolve about a diameter, it will become more and 
more oblate as the velocity of rotation increases, till the 
equatorial diameter have to the polar axis the proportion of 
2*7197 to 1 : arrived at this point the rotatory velocity must 
decrease, in order that the fluid in equilibrio continue to have 
the figure of an ellipsoid of revolution with increasing ob- 
lateness ; in so much that while the oblateness tends to be 
infinite, and the fluid to become a circular sheet in the plane 
of the equator, the velocity of rotation continually approaches 
to zero. 
As the oblateness increases without ceasing, there is but 
one rotatory velocity with which a spheroid of a given figure 
will be in equilibrio. 
But when a fluid mass is to revolve in a given time, and 
the figure that will maintain the equilibrium is sought, there 
are two solutions, if the proposed rotation be within the maxi- 
mum, and one only, when it reaches that limit. 
When the rotatory velocity is greater than the maximum, 
the equilibrium cannot take place : for, on the one hand, the 
proposed rotation is inconsistent with the figure of an ellip- 
soid ; and, on the other, it has been proved, that a homoge- 
