WITHDRAWN FROM THE ACTION OF GRAVITY. 357 
the case of the two preceding figures, to determine the complete form of the 
meridian line. ; 
We will mention first a remarkable transformation which the partial figure 
undergoes when the ratio between the distance and the diameter of the rings is 
sufficiently below § to allow the abstraction of a large quantity of liquid with- 
out occasioning disunion, and we carry this abstraction as far as possible. The 
constricted portion and the bases alike becoming more concave, we know there 
must arrive a moment after which their surfaces can no longer co-exist without 
mutually cutting one another; there is then produced a phenomenon of the same 
nature as with the liquid polyhedrons, (2d series, §§ 31 to 35)—that is to say, the 
figure passes gradually to a laminar state: two conical films are seen to form, 
proceeding respectively from each of the rings, and at the centre of the system 
a plane film, such as is shown in meridian section at Fig. 23. These films ac- 
quiring more and more development in proportion to the continued absorption 
of oil, the whole tends finally to be reduced to a sort of double laminar and 
truncated cone; but one of the films always breaks before we can reach that 
point. It hence results that if we wish to observe the constriction in all its 
phases with the form proper to it as pertaining to the new figure of equilibrium, 
it is necessary to oppose an obstacle to the generation of films. Now this is ac- 
complished without difficulty by substituting disks for rings, and thus prevent- 
ing the bases from becoming concave; we may then remove oil until the figure 
spontaneously disunites at the middle of its height. 
§ 23. Before pursuing the meridian line beyond the limits of the partial figure, 
we should offer two important remarks. 
In the first place, the constricted portion, whether realized between rings or 
disks, always shows itself perfectly symmetrical on both sides of the cercle de 
gorge. This is eqtfally required by the theory, for the mode of reasoning of 
§ 6 is independent of the nature of the meridian line, and applies as well to the 
constricted portion with which we are occupied as to that of the unduloid. If, 
then, in a meridian plane, we imagine a right line perpendicular to the axis of 
revolution and passing by the eentre of the cercle de gorge, all that the com- 
plete meridian line presents on one side of the above right line, it will also pre- 
- sent, in a manner exactly symmetrical, on the other side, so that this right line 
will constitute-an axis of symmetry. 
In the second place, since, by employing rings, the bases of the partial figure 
are concave, it follows that, through the whole extent of the complete figure 
the pressure is less than that of a plane surface. Now, agreeably to the formula 
1 
dio pl 
RR 
eee! ; 
the notation adopted in this series, Moy should be finite and negative. In 
of such pressure, (§ 14,) this requires that the quantity , or, according to 
our new figure, therefore, the mean curvature (2d series, §§ 5 and 6) is negative— 
that is to say, at each point of this figure concave curvatures predominate. 
§ 24. The points @ and 4, (Fig. 22,) at which the partial meridian line stops, 
cannot, in the complete meridian line, be points of inflexion. We see, in fact, 
from the direction of the tangent at those points, that if the meridian line, at 
its departure thence, pursued a curvature in the contrary direction, (Fig. 24,) 
the radius of curvature would, in this part of the figure, be directed to the in- 
1 1 
—+4— l 
MN would 
become positiye ; which cannot be, by reason of what has been said above. 
Beyond the’points a and 4, then, the meridian line begins with a concave cur- 
vature; and the same direction of curvature is evidently maintained, for the 
same reason, so long as the curve contiuues to retire at once from the axis of 
revolution and the axis of symmetry. But the curve cannot continue to sepa- 
terior of theliquid like the perpendicular, and that thus the quantity 
