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WITHDRAWN FROM THE ACTION OF GRAVITY. 353 
must be finite, and indeed cannot be considerable relatively to the diameter of 
the rings.* 
It is evident that, in its transit to the axis of revolution, the curve has passed 
through all the conditions which, with the given rings, are consistent with 
equilibrium ; the above maximum, then, constitutes a limit of separation for the 
rings, beyond which there can be no catenoid between them. 
But the preceding furnishes another consequence equally remarkable. Since, 
during the transit of the summit of the catena, the points A and B first with- 
draw from and afterwards again approach one another, they necessarily repass 
by the same distances, so that, for each distance less than the limit, they per- 
tain at once to two catene. Now, it results from this that, to every degree of 
separation less than the maximum, there always correspond two distinct cate- 
noids resting on these rings, but penetrating unequally between them. We see 
without difficulty that the summits of the two generating catenz, summits 
which, for a separation null, are the one at the common periphery of the rings 
in contact, and the other on the axis of revolution, approach one another more 
and more in proportion as the separation increases, and finally coincide, equally 
with the two entire curves, when that separation attains its maximum. ‘Thus 
the two catenoids will differ so much the less as the separation of the rings is 
greater, and when the limit has been reached will form but one. 
§ 17. All catenz are, we know, alike; and hence if we imagine a series of 
complete catenoids generated by catenz of different dimensions, all these catenz, 
‘from the condition which they must satisfy, (§ 14,) will be similarly placed in 
relation to the axis of revolution, and consequently all the catenoids will be 
similar figures. The complete catenoid, then, is not susceptible of variations 
of form like the unduloid, but constitutes an unique figure, like the sphere and 
the cylinder. Hence the two complete catenoids whigh, theoretically, rest on 
the same rings, when the separation of these is below the limit, do not differ 
from one another except by their dimensions absolutely homologous. 
§ 18. Of the two partial catenoids pertaining to these two complete catenoids, 
and equally possible by the theory between the rings, our process necessarily 
gives that which is least re-entering; if we attempt, by removing further 
quantities of oil from the mass, to realize the most deeply re-entering catenoid, 
there is always, as we shall presently see, another figure of equilibrium pro- 
duced. From the impossibility, therefore, of realizing this partial and most 
deeply re-entering catenoid, we may justly conclude that it would constitute an 
‘unstable figure of equilibrium. 
As to that which is least re-entering, it evidently forms a portion of the com- 
plete catenoid, so much the more extended as the separation of the rings is 
nearer its maximum; for, in proportion as the rings are more widely separated, 
the are of the catena which they intercept between them is (§ 16) a more con- 
siderable portion of the curve. In order to have a partial catenoid more ex- 
tended in relation to the complete catenoid, it would be necessary that the 
catena should penetrate more deeply between the rings; but then, by however 
small a quantity the summit of the curve should advance, the separation of the 
rings would diminish, (7bid.,) there would be another catena possible, less re- 
entering and resting on the same rings, and the partial catenoid generated by 
the first catena being the most re-entering, it would be unstable. The catenoid 
of greatest height constitutes, then, the most extended portion of the complete 
eatenoid which can be realized between two equal rings. 
We will notice here another consequence to which the above would seem to 
lead, and which would yet be opposed to fact; for every degree of separation 
* We can determine its precise value by means of the equation of generating catene, but 
this calculation is reserved for the series in which we shall unite all the applications of mathe- 
matical analysis with the subject of our researches. 
23 8s 
