WITHDRAWN FROM THE ACTION OF GRAVITY. 365 
know that the convex partial figure with which we started is not portion of an 
unduloid, since its convexity exceeds the sphere; it is perceptible that it is 
not portion of a catenoid, and from what precedes we see that the above con- 
traction cannot be portion of a node. 
Thus our original hypothesis of a point of inflexion in the part of the curve 
which is withdrawing from the axis of symmetry and approaching the axis of 
revolution leads inevitably to impossibilities, and, consequently, the curve 
maintains the same direction of curvature until it deviates from those conditions. 
But to do so it is evidently necessary that it should first cease to withdraw 
from the axis of symmetry, or, in other terms, that it should present a point 
where the tangent is parallel to that axis. Neither is this point one of inflexion, 
for the perpendicular and the radius of curvature would there be both infinite, 
La 
which would annul the. quantity MON Beyond this point, then, the curve 
redescends towards the axis of symmetry, still preserving the direction of its 
curvature. Further, this same direction is maintained, as we shall show, so long 
as the curve continues to descend. In effect the liquid of the partial figure 
realized, and which has served us for a point of departure, being situated in the 
concavity of the curve, we readily see that at all the points of our descending 
branch the perpendicular is negative. But if this branch contained a point of 
N ? 
and consequently, on account of the sign of the perpendicular, would be also 
negative; while on the meridian are of the realized partial figure the radius of 
1 
inflexion the quantity MN would be reduced at that point to the term = 
it 
curvature and the perpendicular being both positive, the quantity Me N® itself 
positive. 
But the branch in question cannot descend indefinitely by still approaching 
the axis of revolution, or, in other terms, cannot tend toward an asymptote par- 
allel to that axis; for, at the point situated at infinity on the asymptote, the 
i | 1 
quantity MIN would again be reduced to the term y’ and consequevtly would 
be again negative ; it is necessary, therefore, that one branch should pass at a 
minimum of distance from the axis of revolution, and should thus form the 
generating arc of a constriction; and as this constricted portion could pertain 
neither to the unduloid nor the catenoid, it necessarily constitutes the summit 
of a node of the nodoid. We must recur, therefore, to the meridian line of the 
nodoid, and conclude that all the figures obtained in the experiments of § 31 
are partial nodoids, whatever the degree of approximation of the disks, pro- 
vided the spherical curvature be overpassed, and whatever the volume of oil in 
relation to the diameter of the disks. 
§ 34. We are now in a position to consider what is the nature and what the 
limits of the variations of the nodoid. Since, in the experiments of § 31, we 
pass by a portion of a sphere, after which, as has been just seen, the partial 
nodoid is immediately realized, and since the latter then varies continually until 
it reaches the phase at which instability commences, it is obvious that the por- 
tion of a sphere constitutes one of the limits of these variations, and that hence 
the limit of the corresponding variations of the complete nodoid is an inde- 
finite series of equal spheres, having their centres on the axis. But it will 
readily be perceived that the only possible mode of continuous variation « 
tending towards that limit is the following : in proportion as the complete nodoid 
approaches the series of spheres, the dimensions of the nodes as well as the 
distance of their summits from the axis diminish more and more, while the cur- 
vature of the ares which connect these nodes verges towards that of the cir- 
