354 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
less than the maximum, the catenoid which is least re-entering always proves 
perfectly stable, and, as has been shown above, that which is most re-entering 
must be regarded as always unstable. Now, the catenoid of greatest height 
forms, as has also been seen, the transition between the catenoids of the first 
category and those of the second, and consequently between stable and un- 
stable catenoids ; we might, therefore, take it for granted that the catenoid of 
greatest height is.at the limit of stability of that kind of figure; and yet, when 
we realize it with a mass of oil, it manifests a decided instability. We shall 
presently know to what this apparent contradiction is attributable. | 
§ 19. It is readily seen that the third limit of the variations of the unduloid, 
a limit spoken of in § 9, is nothing else but the catenoid. In effect, by causing 
the partial unduloid to vary in the manner indicated in that paragraph, it is 
clear that, in proportion as the volume of the mass is increased, the perpen- 
dicular and the. radius of curvature relative to the summit of the convex meri- 
dian are continue to increase and become infinite at the same time with the 
ft ad ; 
—-+—is null, and this 
volume; whence it follows that at that limit the quantity MN 
we know to be the character of the catenoid. 
N 
zero in proportion as the unduloid approaches the catenoid, the general ex- 
pression (§ 14) of the pressure exerted by an element of the superficial stratum 
shows that this pressure tends, at the same time, towards that of a plane sur- 
face. If, then, we imagine between two rings a constricted portion pertaining 
to an unduloid, and that this unduloid is tending, by degrees, towards a cate- 
noid, the bases of the figure, bases whose pressure must always be equal to 
that of the constricted portion, will necessarily become less and less convex, 
and be finally altogether plane. Now, this is what is evidently realized by the 
experiment of § 14; when, after having formed between two rings a cylinder 
whose height does not exceed % of the diameter, we gradually withdraw liquid 
from it and the bases sink, by degrees, till they lose all curvature, the constric- 
tion which is produced and which deepens in the same proportion pertains to 
an unduloid which is tending towards its third limit, and thus the experiment 
in question exhibits before our eyes the progressive transition of the unduloid 
into the catenoid. If we collate the preceding with the contents of § 13, we 
shall be authorized to deduce the conclusion that every constriction, resting on 
two rings and presenting convex bases, is a constriction of an unduloid, whether 
the curvature of the bases be superior, equal or inferior, to that of the bases of 
the cylinder which would be comprised between the same rings. 
§ 20. We will recite, now, the cixeumstances which have been presented to 
us by the experimental investigation of the partial catenoid of greatest height. 
The diameter of the rings employed was 71 millimetres. In all the experi- 
ments which follow, the process commenced with forming a cylinder, and then 
oil was withdrawn from the mass, at first by the syringeful and afterwards by 
small portions; from time to time the operation was suspended in order to 
observe the figure. 
First experiment.—Distance of the rings 55 millimetres. The versed sine 
of the spherical caps, which constitute the bases, is gradually reduced to a frac- 
tion of a millimetre ; then, during an interruption of the exhaustion, a singular 
phenomenon is produced; the figure undergoes a slight spontaneous modifica- 
tion; the convexity of the bases rapidly augments until the versed sine re- 
trieves a value of about 2.5 millimetres, and consequently the constriction 
formed between the rings becomes somewhat thinner, and then the whole re- 
mains stationary, By still cautiously absorbing oil, the versed sine increases 
: ibe | 1 FIED 
The quantity MTN’ or, what is the same thing, RR’ tending thus towards 
