200 M.G. BR. Dahlander on the Equilibrium of a Fluid Mass 
—Mm+M"a 7 
—2= ME Mp | 
—Mn+M"6 
cer —M+M!—M! 4 w?’ | (4) 
—Np+N'y 2? f 
= ae M! + yw? am | 
a —N+N!—N! | 
2 —M+M RM yur | 
From which we obtain 
—M'm 7 
*=M—M—u? | 
—M!n 5 
cow Te 
1 EN och | 
Y= N—N! J 
From the first two of equations (5) it follows that 
2.2 Batti 
The geometrical signification of this proportion is, that the 
centre of the fluid mass be in the plane which passes through 
the axis of rotation and the centre of the imner bounding surface 
of the spheroid. We find, moreover, that the position of this 
centre in a state of equilibrium with a fixed angular velocity is 
independent of the density of the fluid, supposing the form and 
density of the outer spheroid to be the same. Further, we find 
that if a fluid ellipsoid satisfy the conditions of equilibrium, all 
other similar ellipsoids of the same density will also satisfy these 
conditions. Generally, from equations (5), real finite positive or 
negative values of a, 6, y can be obtained if M, M’, N, N!, m, n, 
p, and ware given. But one or more of these values may become 
infinite under certain circumstances. This is the case when the 
two bounding surfaces of the solid spheroid are similar. Then 
M=WM! and fp N’, whence the value fcr y would be infinite, 
unless at the same time p=O0, m which case y can have any 
value whatever. 
The last of equations (4) constitutes the real equation of con- 
dition, which determines the relation which must subsist between 
the density, form, and angular velocity of the fluid mass, in order 
that equilibrium may be possible for the given values of M, M’, 
N, and N’. We shall separately consider the particular case when 
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