WITHDRAWN FROM THE ACTION OF GRAVITY. 345) 
quantity increases while the other decreases, it cannot preserve the same value, 
and equilibrium is impossible. If we suppose, on the contrary, that the curva- 
ture of the are a@ m, on parting from a, diminishes more than that of the arc a x, 
we shall conclude, by the same mode of reasoning, that the quantity its 
) 
would likewise change its value in passing from one of the parts of the curve 
to the other. 
Thus the hypothesis of curvatures either greater or less in the are a m than 
in the are a@ 7 is incompatible with the equation of equilibrium; it is conse- 
quently necessary, in order to satisfy this equation, that, on the small prolonga- 
tion a m, the curvatures should be identically the same as on an are a x of the 
same length taken on the other side of a. Now, it is clear that this implies the 
identity of the whole portion of the curve situated beyond the point a with the 
portion situated within it. The portion of the curve comprised between @ and & 
(Figs. 6 and 7) will be reproduced, therefore, beyond a, and, for the same reasons, 
will be still reproduced indefinitely; and the same will be the case on the other 
side of the point 4, in such manner that the meridian line will represent an un- 
dulating curve extending to infinity along the axis, approaching and retiring 
alternately and periodically by equal quantities. 
The complete figure of equilibrium, therefore, is prolonged to infinity along 
the axis, and is composed of a regular and equal succession of expanded and 
constricted portions, of which Fig. 9 represents a meridian section of a certain 
Fig.9 
extent. To this figure of equilibrium we shall apply, in the sequel, the name 
of wnduloid, from the form of its meridian line. 
§ 7. Itis easy to conceive how equilibrium may exist in such a figure, although 
in the dilated parts the curvature is convex in all directions around the same 
point, while, in the constricted parts, the curyature is convex in certain direc- 
tions and concave in others: it is because, in these latter parts, the convex or 
positive curvatures are stronger than the concave or negative curvatures, so that 
the mean at each point (2d series, § 6) is positive and equal to that which cor- 
responds to the different points of the dilated parts. From the fact that, 
in the unduloid, the mean curvature is positive, it necessarily results that when- 
ever we realize any portion of an unduloid between two rings, the bases which 
rest upon the latter will be convex spherical caps. 
§ 8. If, in the experiment of § 4, the volume of oil remaining the same, we 
employ a solid cylinder of greater diameter, the liquid mass extends still more 
in the direction of the axis, and the meridian curvatures diminish, so that, in 
the corresponding complete figure, the expansion and constrictions are less de- 
cided. ‘Thus the meridian curvatures, in the partial and consequently in the 
complete figure, become proportionably effaced as the diameter of the solid cylin-, 
der is greater; whence we perceive that, in these variations, the complete figure 
tends towards the cylindrical form, which may be considered, therefore, as the 
limit of the variations. 
If, on the contrary, while the volume of oil still remains the same, we employ 
a solid cylinder of smaller diameter, the liquid mass becomes constricted in the 
direction of the axis, the meridian curvatures augment, and the figure approxi- 
mates more and more to the sphere; thus, for instance, when for a mass of oil 
constituting primarily a sphere 6 millimetres in diameter, we take, as solid cylin- | 
