358 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
rate indefinitely from those two axes: in effect, if such were its course, it is 
clear that the curvature must diminish so as to be annulled, in each of the two 
branches, at the point situated at infinity, whence at that point the radius of 
curvature would have an infinite value; and as it would evidently be the same 
: Eaeaey 
as regards the perpendicular, the quantity MTN would become null at that limit. 
It necessarily follows that at a finite distance from its summit the curve has two 
points in which its elements are parallel to the axis of symmetry, and this ex- 
periment confirms, as we are about to see. 
§ 25. If we use disks, which are placed ata distance equal to about the third . 
of their diameter, and carry the absorption of liquid sufficiently far, the angle 
comprised between the last elements of the surface of the mass and the plane 
of each of the disks diminishes until completley annulled, so that that surface is 
then tangent to the planes of the disks, (Fig. 25,) and hence the last elements 
of the meridian line are parallel to the axis of symmetry. It is very difficult 
to judge of the precise point where this result is attained, but we ascertain that 
it is really produced by continuing the exhaustion of the liquid: we soon see 
the circumferences which terminate the surface of the mass abandon the mar- 
gins of the disks, withdraw, by a diminution of diameter, to a certain distance 
within them, and leave a smal! zone of each of the solid planes free; now, as 
these zones remain necessarily moistened with oil, though the stratum be ex- 
cessively thin, it is clear that the surface of the mass must there meet the’ planes 
tangentially. If the separation of the disks is still less, we obtain a result of 
the same nature; only, before spontaneous disunion takes place in the middle 
of the figure, we may still further contract the circumferencesof contact, or, in 
other words, enlarge the extent of the free zones. ‘ 
§ 26. The reason assigned in § 24 to establish the absence of an inversion 
of curvature so long as the curve withdraws at once from the axis of revolu- 
tion and the axis of symmetry, evidently still holds good at the points which 
we have just been considering, that is to say, at those where the elements are 
parallel to this last axis; whence it follows thatthe curve afterwards approaches 
this latter axis, by preserving the same direction of curvature, as is shown at 
Fig. 27, where the curve is drawn on a larger scale than the portion comprised 
in Fig. 25, and where the axis of symmetry is represented by the right line XX’. 
And so long as these prolongations of the curve continue to withdraw from the 
axis of revolution, the direction of the curvature must still remain the same. 
For let us suppose that it changes, at f and at g for instance, (Fig. 28,) then, 
from the point f to a point such as m, situated a little beyond, the radius of 
curvature and the perpendicular would have, it will be seen, opposite directions, 
el: 
so that the quantity —+ — would be a difference; now, from f to m the per- 
ig Sh aL : 
pendicular would evidently go on increasing, since, on one hand, the distance 
from the axis of revolution increases, and, on the other, that perpendicular would 
have a still greater and greater obliquity ; it would, therefore, be necessary, in 
order for the above difference to remain constant, that the radius of curvature 
should also continue to increase from f to m; but this is precisely the contrary 
of what would occur, for, by reason of the inflexion, the radius of curvature 
would be infinite at f, and consequently could only diminish after leaving that 
point. It is needless to remark that what has been just said applies equally 
to the point g. 
Let us see now whether, before reaching the axis of symmetry, the curve 
can present two points, such as / and k, (Fig. 29,) where its el@ments shall be 
perpendicular to that axis. With that view we will examine what conditions 
the curvature should satisfy from the summits to the points 2 and 4, and it 
will suffice to consider the are sx. Let m be the point where the element of 
