WITHDRAWN FROM THE ACTION OF GRAVITY 369 
this it results that the curvature could nowhere change its direction: for if 
there were a point of inflexion, the equation of equilibrium would be there also 
ft 
reduced to no: and consequently the perpendiculars at the above first ex- 
treme point, and at the point of inflexion, would be equal, which is evidently 
impossible. Therefore, the curve being free from all undulation, the curvature 
would necessarily tend towards zero, or, what amounts to the same, the radius 
of curvature would tend towards infinity in approaching the second extreme 
point, so that at that point the term iM would disappear as at the former, which 
would require, as before, the impossible equality of the two perpendiculars. 
The sole figures, therefore, of equilibrium of revolution of a liquid mass with- 
drawn from the action of gravity are those at which we have arrived in the 
second and in the present series, namely: the sphere, the plane, the cylinder, 
the unduloid, the catenoid, and the nodoid. All these figures, with the excep- 
tion of the sphere, having infinite dimensions in certain directions, it results 
that, among the figures of equilibrium of revolution, it is only the sphere which 
can be realized in a complete state witha finite mass of liquid; hence, as we 
have seen, it is always the spherical form which is assumed by a mass of oil 
abandoned to itself in our alcoholic mixture. 
ff 
[TO BE CONTINUED IN THE NEXT REPORT.] 
24 8 
