we 
352 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
of Fig. 19, in which the liquid fills the space left by the catena in revolving 
between itself and the axis, and another in which the liquid occupies the space 
embraced by the curve. A meridian section of the latter is represented by 
Fig. 20. 
: 16. In the experiment of § 14 we only succeed, as has been said, in ren- 
dering the bases of the figure plane when the separation of the rings does not 
exceed about 3 of their diameter. We shall recur, further on, to the details of 
this experiment, which presents some curious particulars; but there is an im- 
portant consequence which is deduced immediately from it, and which requires 
our notice at present; for rings of a given diameter there is a maximum of 
separation beyond which no portion of a catenoid is any longer possible be- 
tween them. We shall proceed to show that this result is in accordance with 
the theory, and we shall, at the same time, be conducted to a new result. 
We have seen that the generating catena should satisfy the condition that 
the radius of curvature of its summit be equal and opposite to the right line 
which measures the distance of that summit from the axis of revolution. ‘This 
being so, let us conceive, in a meridian plane, a right line perpendicular to the 
axis of revolution, and representing the axis of symmetry of the catena, and 
again a second right line parallel to the axis of revolution and distant from the 
latter by a quantity equal to the radius of the rings. Let us conceive, further, 
in the same plane, a generating catena having its summit at the point where 
the right line of the rings is intersected by the axis of symmetry of which we 
have spoken. This catena will be tangent at that point to the right line in 
question, and consequently cannot rest upon the rings except when the peri- 
pheries of these pass by the point of tangence, or, in other words, when the 
mutual distance of the two rings shall be null;* the catena under consideration 
corresponds then to the case of a separation null of the rings. Let us now 
suppose that the curve quits this position, and proceeds gradually towards the 
axis of revolution, being so modified as always to satisfy the condition of 
equality between the radius of curvature of its surhmit and the distance of that 
summit from the axis; in each of its new positions it will cut the right line of 
the rings at two points, which we will designate as A and B. The distance of 
these two points will then represent, in each of these positions, the distance 
apart of the rings, and the corresponding catena will represent the meridian 
line of a catenoid, of which the rings would comprise a portion between them. 
This being premised, let us consider the evolutions of the points A and B. In 
the initial position of the catena, when its summit is tangent to the right line 
of the rings, these points are confounded at the point of tangence; but when 
the summit of the curve begins to advance towards the axis of revolution, they 
separate and progressively remove from one another. But their mutual dis- 
tance will attain a maximum, after which that distance will continue to diminish. 
In effect, conformably With the condition attached to the catena, when its sum- 
mit shall have arrived very near the axis of revolution, the radius of curvature 
of that summit will have become very small; whence it follows that the two 
branches of the curve will closely approach one another, and that, conse- 
quently, the two points A and B will also be in close proximity ; finally, when 
the summit is on the axis, # eso same points will be again reunited, as then the 
radius of curvature of the summit will be null, and the two branches of the 
curve will form but a single right line coincident with the axis of symmetry. 
Thus the points A and B, which were first coincident and then diverged from 
one another, afterwards approach, until at last they again coincide; from which 
it necessarily follows, as just stated, that their mutual distance attains a maxi- 
mum; and it is easy to see, from the nature of the curve, that this maximum 
* For simplicity, we here consider the two rings as possessing no thickness. 
