WITHDRAWN FROM THE ACTION OF GRAVITY. 341 
and, on the other hand, the perpendicular at m will be more oblique to the axis 
than the radius drawn from; for this double reason the perpendicular in ques- 
tion will be greater than the radius of the are of the circle; but, because of the 
inferiority of the curvature at m, the radius of curvature at that point will be 
also greater than the radius of the are of the circle. 
From all this it resuits that the values of M and N, corresponding to the 
point m, are both superior to those which correspond to the point d; but it is 
clear that M and N are of the same sign throughout the length of the are a 6 d, 
and that thus, at the point m as at the point d, the quantity _ 
a sum; this same quantity, then, is smaller at m than at d, and, consequently, 
the equilibrium of the liquid figure generated is impossible. 
If we suppose, now, that the curvature of our meridian are constantly dimin- 
ishes, as is seen in a’ 8! d’, (Fig. 2,) it is apparent that then this arc will be 
interior to the arc of a circle a’ c’ d’, having its centre on the axis, that its cur- 
vature will at first be superior and become afterwards inferior to that of the 
latter, and that at the point d' one of the arcs will again intersect and not be a 
tangent to the other; whence we may conclude, by the mode of reasoning 
constitutes 
employed in the preceding case, that the quantity eee is greater at a point 
near a’ than at d’, so that the figure generated is, as before, impossible. 
Hence, when the meridian line meets the axis, the condition of equilibrium 
cannot be satisfied unless that line is a circumference of a circle, having its 
centre on the axis, or, if we suppose the radius of this circumference to be infi- 
nite, a right line perpendicular to the axis; the figure generated, therefore, is 
necessarily either a sphere or a plane. 
From this flows, as a necessary consequence, the truth of the proposition 
which I advanced (2d series, § 28) from the results of experiment, namely, that 
when a continuous and finite portion of a surface of equilibrium rests on a cir- 
cular periphery, that portion must constitute a spherical cap or a plane. To 
be otherwise, it would be necessary that the cap should not be a curve of revo- 
‘ lution—a supposition which is never realized. 
§ 3. The meridian lines of such other figures of equilibrium of revolution as 
ean have no point in common with the axis, must either be extended infinitely, 
or be closed beyond the axis.. The first class will generate figures which 
extend to infinity, and of these the cylinder has already afforded an example. 
The second would yield annular figures; and we shall see, at the end of the 
present series, whether the existence of figures of that kind is possible. 
To simplify the investigation of the lines in question, we shall proceed to 
demonstrate that they contain no point of retrogression. If we suppose the 
existence of a point of that nature, there are three cases to be considered : first, 
that in which the tangent at the point of retrogression, a tangent which is com- 
mon to the two branches of the curve, is not perpendicular to the axis of revo- 
lution, whatever direction it may otherwise have; second, that in which this 
common tangent is perpendicular to the axis, and where the two branches ap- 
proach the latter in proceeding towards the point of retrogression ; and, third, 
that in which the common tangent, being again perpendicular to the axis, the 
two branches, in proceeding towards the point of retrogression, withdraw from 
that axis. / 
First case-—By casting the eyes on Fig. 3, which represents, in meridian 
sections, portions of the liquid figure, for different positions of the point of retro- 
ression in relation to the axis of revolution ZZ’, we readily perceive that in 
the neighborhood of that point the perpendicular is always, as regards one of 
the branches, directed to the interior of the liquid, and is consequently positive ; 
