WITHDRAWN FROM THE ACTION OF GRAVITY. 351 
, BN) ey 
stratum has for its value Patty Gt 3h) an expression in which A is a con- 
stant dependent on the nature of the liquid and which cannot be null, and P the 
pressure corresponding to a plane surface. Now, in the case with which we 
are occupied, the pressure at any point of the complete figure must be equal to 
that of a plane surface, since the bases of our partial figure are planes; the 
above expression then will, in this case, be reduced to P, so that we have 
RO SneR | 
RR 
face the mean curvature (2d series, §§ 5 and 6) is null, or, in other terms, at 
each of these points there are, as in the portion formed between our rings, con- 
cave curvatures whose effect exactly destroys that of the convex curvatures, so 
that the pressure remains the same as if there had been no curvature. 
0. Thus the figure in question is such that at each point of its sur- 
vaveon Dal ah ; ; a! , 
Now, the equation RR? becoming here, according to the notation which 
R 
Reg] 
we have adopted for figures of revolution, Mon” we deduce therefrom 
M=—=—N;; whence we see that, at each point of the meridian line, the radius 
of curvature is equal and opposed to the perpendicular. Now, geometers have 
demonstrated that the only curve which possesses this property is the catena, 
(chainetie.)* 'This, then, is so placed relatively to the axis to which the per- 
pendiculars are referred, that the right line, which divides it symmetrically into 
two equal parts, shall be perpendicular to that axis, and the summit of the 
curve distant from the point of intersection of those two right lines by a quan- 
tity equal to the radius of curvature of that summit. Our figure, then, in its 
complete state, is that which would be generated by the revolution of a catena 
thus placed in relation to the axis. We will, accordingly, give it the name of 
catenoid, of which Fig. 19 represents a meridian section sufficiently extended, 
the axis of revolution being ZZ. 
The catena being a curve, whose branches are infinite, the catenoid also is 
extended to infinity, like the cylinder and the unduloid, but no longer in the 
direction only of the axis. 
§ 15. We recall here a principle which was cursorily noticed in § 8 of the 
2d series, and of which we afterwards made use in § 31 of the same series: 
when a: surface satisfies the general condition of equilibrium of our liquid 
figures, that condition is equally fulfilled whether we suppose the liquid on one 
or the other side of the surface in question. In effect, the inversion of the posi- 
tion of the liquid, with regard to the surface, only changes the signs of the two 
principal radii of curvature corresponding to each of the points of the latter, 
but evidently does not at all alter the absolute values of those radii, so that if 
ate is constant in one of the cases, it will be so in the other. 
There are always, then, for any one surface which satisfies the condition of 
equilibrium, two liquid figures, the second of which, presents in concave what 
the other presents in relief, and, vice versa, figures*which are both figures of 
equilibrium. We see this realized, for instance, in our experiments as regards 
the sphere; a mass of oil left free to itself in the midst of the alcoholic mix- 
ture gives a sphere in relief, and, on the other hand, when some of the alcoholie 
mixture is introduced into one of our masses of oil, the surfaces into which the 
bubbles of this mixture are moulded constitute spheres of oil in concave, (2d 
series, § 10.) In virtue of this principle we have two catenoids; that, namely, 
the quantity 
* The catena will be recognized as the curve formed, in a state of equilibrium, by a heavy 
and perfectly flexible chain suspended at two fixed points. es 
. 
