364 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
equally with the unduloid, susceptible of variations between certain limits, Fig. 
38 should be regarded only as presenting one case of its meridian line. We 
LeY. o8 
will? urther recall the observation made in § 29, and which will now be better 
understood from the appearance of this curve, namely, that the complete figure 
can only be represented in the state of a simple surface, since, if it were supposed 
to be full, there would evidently be parts of it engaged in the mass. 
§ 33. Before we proceed to the consideration of the nodoid in its variations, 
a question should be resolved which is suggested by the experiments of § 31. 
Since we know now the form of the meridian line, we see that those experiments 
realize the portion of the nodoid generated by a part, more or less considerable, 
of one of the ares convex towards the exterior, such as xpn’, (Fig. 38.) 
But it may be asked if this does not require that, with disks of a given diameter, 
the volume of oil should be comprised within certain limits, so that for larger or 
smaller volumes the figure realized would no longer pertain to the nodoid. To 
determine this, let us take one of the figures realized, follow the meridian are 
beyond the point where it meets the edge of one of the disks—the upper one, for 
instance—and let us see whether it be possible to arrive at a curve other than 
the meridian line of a nodoid. 
We will suppose, first, that in that part of its course where it continues to 
approach the axis of revolution, and to withdraw from the axis of symmetry, 
the curve presents a point of inflexion, so that it shall afterwards turn its con- 
vexity towards those two axes. If, while still approaching the first, it changed 
a second time the direction of its curvature, the perpendicular corresponding to 
this second point of inflexion would necessarily be shorter than the perpendicular 
corresponding to the first, since it would have less obliquity, and would proceed 
from a point nearer the axis. But this is incompatible with the equation of 
equilibrium; for this equation being reduced at all the points of inflexion to 
1 
N =O, the two above perpendicwlars must be equal. The existence of this 
second point of inflexion being thus impossible, we see that beyond the first, the 
curve, which cannot (§ 2) attain the axis of revolution, must necessarily either 
tend towards an asymptote parallel to that axis, or else present at a finite dis- 
tance a point where the tangent shall be parallel to the same axis. 
That the first of these two conditions must be rejected is at once obvious; 
for at the extreme point where the curve would touch the asymptote the radius 
of curvature would be infinite, which would again reduce the equation of 
1 
equilibrium at that point to noo and the perpendicular would there also be 
evidently shorter than at the point of inflexion. In the second case, the point 
where the tangent would become parallel to the axis of revolution cannot, on 
account of the evident inequality of the perpendiculars, be a second point of 
inflexion. It would then constitute a minimum of distance to the axis, and con- 
sequently a small are extending on both sides of this minimum would generate 
a constriction which might be realized between two equal rings or disks. Now 
we have discussed all the possible partial figures of that nature. We have 
seen that every constricted portion pertains either to an unduloid or a catenoid, 
or to the part of a nodoid, which encompasses the summit of a nodus; but we 
