362 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
equal to that of the are ay. The point 7 will be more remote from the axis than 
the point #; and moreover, on account of the inferiority of the curvatures, the 
tangent at 7 willform, with this axis, a greater angle than the tangent at 7; the 
perpendicular, therefore, at the point 7 will, for this double reason, be longer 
than the perpendicular at the point 8. Again, and still by reason of the in- 
feriority of curvatures, the radius of curvature at the point 7 will also be longer 
than the radius of curvature at the point #; but, at this latter point, these two 
quantities have the same value as at the point a. In passing then from.a to 7, 
the radius of curvature and the perpendicular will both increase. Now this is 
incompatible with the equation of equilibrium; as the curve, throughout the part 
which we are considering, turns its concavity towards the axis, the radius of 
curvature and the perpendicular have everywhere the same sign, and conse- 
quently when one increases the other should diminish, and vice versd. If we 
suppose that, at parting from a, the curvature goes on increasing, the are of the 
curve will be interior to the are of a circle, and the same mode of reasoning 
would enable us to see that from one to the other extremity of the former the 
radius of curvature and the perpendicular will both diminish. The hypothesis 
of an asymptote parallel to the axis of revolution leading thus to an impossible 
result, we see that it must be rejected like the first. 
It is the third hypothesis, therefore, which is true; that is to say, the curve 
presents two points, p and p’, (Fig. 33,) where the tangent is parallel to the axis 
of revolution. ; 
§ 31. Experiment fully confirms this theoretical deduction, and furnishes, be- 
sides, a suggestion for the discovery of the ulterior course of the curve. 
The two disks being placed at any distance from one another—a distance, 
for instance, equal to their diameter—we form a cylinder between them, and 
then gradually lower the upper disk: the figure then passes, we know, to the 
unduloid, and swells more and more till it constitutes portion of a sphere, (Fig. 
34.) But if we continue to lower the upper disk, the meridian convexity still 
augments, and consequently passes beyond the above point; for a certain ap- 
proximation of the disks, we thus obtain, for example, the result represented by 
Fig. 35, and the liquid figure is always perfectly stable. Now, in this state, it 
ean no longer form part of an unduloid, since the sphere has been exceeded, 
which is one of the limits of the variations of that figure, (§ 8.) We may 
again lower the disk until, at the points where the meridian line reaches the 
borders of the disks, the tangents shall be nearly perpendicular to the axis of 
revolution, as is seen in Fig. 36, and for a less mass of oil in Fig. 37. It is 
even possible that perpendicularity may be attained ; but it would be very diffi- ~ 
cult to acquire the assurance of this, because, on the one hand, the eye cannot 
judge with sufficient precision of the direction of these extreme tangents, and, 
on the other, the liquid figure, at this degree of approximation of the disks, 
loses its stability; if we depress a little too much the upper disk, the oil is 
observed to be transferred in greater mass to one side of the axis of the system, 
so that the figure ceases to be one of revolution; then, on this same side, the 
oil overflows the borders of the disks, and spreads in part on their exterior 
faces. 
Fig. 34 Pig. 33 Fig, 36 Tig. 37 
eS 
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_ 
Now, in virtue of what has been stated in the preceding paragraph, so long 
as the curve, at parting from m, (Fig. 33,) continues to withdraw from the axis 
of revolution, the radius of curvature eannot become equal to the perpendicular, 
and since it is inferior to it at , must remain inferior to it so long as the point 
ae, 
