WITHDRAWN FROM THE ACTION OF GRAVITY. 363 
pis not attained; in the whole extent, then, of the are » ~ p, except at the point 
n, and perhaps at the point p, to which the demonstration does not extend, the 
centre of curvature is always situated between the curve and the axis, and con- 
sequently the curvature is always stronger than that of a circumference of a 
circle having its centre on the axis. But, as we have just seen, in the partial 
liquid figures represented by Figs. 35, 36, and 37, the meridian curvatures are 
stronger than when the figure is a portion of a sphere, or, in other words, 
stronger than that of a circumference of a circle passing by the borders of the 
disks and having its centre on the axis. From this it is clear that these partial 
figures constitute portions of the complete figure generated by an are of the 
meridian linc extending on both sides of the point p, (Fig. 33 ;) only they re- 
late evidently to different-cases of that complete figure which we know to be 
susceptible of variations like the unduloid. 
§ 32. We will take one more step in pursuit of our meridian line. In the 
above experiments, when the densities of the two liquids are rendered quite 
equal, the oil figure is always perfectly symmetrical in relation to its equatorial 
circle. It is by the eye, indeed, that we thus judge, and it might be supposed, 
perhaps, that this symmetry is but approximate; but we shall proceed to 
show that it is exact. In the absence of all accidental cause of irregularity, 
there would be evidently no reason why an excess of curvatures should exist 
rather on one definite side of the equator than the other, since the two disks 
_ are equal and parallel ; whence it results that there is necessarily a form of 
equilibrium in which the symmetry is perfect. But if, in the partial figures 
realized—figures which are stable—symmetry were but approximated, it would 
be necessary to admit that the exactly symmetrical form of equilibrium just 
spoken of would be unstable. If, then, all the liquid figures which can be 
obtained in the experiments described above, that is to say, in those which give 
all the degrees of depression of the disk from the case of Fig. 34 to that of 
Fig. 36, and all the masses greater and smaller with the same disks—if, I say, 
all these figures were symmetrical only in appearance, there would correspond 
to each of them another figure of equilibrium differing extremely little, and 
which would be unstable. Now, the existence of two partial figures of equi- 
librium extremely near, the one stable and the other unstable, may well occur 
in a particular case of the variations of two complete figures, or, at least, of one 
of them, and we have seen an example (§§ 20 and 21) in regard to the contrac- 
tion of an unduloid, when that contraction closely approximates the partial 
catenoid of greatest height; but we can comprehend that it is impossible for the 
same thing to be reproduced in the whole extent of the variations of the partial 
figure realized. Hence we conclude that, in the liquid figures of the preceding 
paragraphs, the symmetry is real, and that, in one complete meridian line, there 
is thus, besides the axis of symmetry of the node, another axis of symmetry 
equally perpendicular to the axis of revolution, and passing by the. point p, 
(Fig. 33.) Consequently, all that the curve presents on one side of this point, 
it should present symmetrically on the other; the node which exists above p 
must have its corresponding node below, and since the two have respectively 
their axis of symmetry, it necessarily results that, in the first place, they are 
perfectly identical, and, in the second place, that all that is found on one side 
of one of them must be identically reproduced on the other side; whence it 
follows that above the upper node there is another like it, and above the last 
still another, and so on indefinitely along the axis of revolution, while the same 
thing occurs below the inferior node, all being connected by ares equally identi- 
eal with one another. An extended portion of this curve is represented at Fig. 
38, in which the axis of ‘revolution A B is placed horizontally. 
The figure generated by this curve is thus prolonged indefinitely in the 
direction of the axis, like the cylinder and unduloid. We will give to this also 
a name, and will call it the nodoid. It should be observed that this figure being, 
