412 THE FIGUEES OF EQUILIBEIUM OF A LIQUID MASS 



tlie figure at the time wlien the two concave bases approach so as nearly to touch 

 one another with their summits, and the two others begin to show themselves 

 when the bases become tangents, along their borders, to the portion of surface 

 comprised between the rings ; the first of these films is plane, 

 ^ig- 1 the rest appear conic. Fig. 1 represents a meridian section of 



the system, in which the films have already acquired a certain 

 development, the dotted lines being sections of the planes of the 

 rings. The films afterwards become more and more extended 

 by the cautious abstraction of new quantities of oil, and the whole tends, as a 

 final result, to give a laminar figure approaching to two truncated cones united 

 by their small bases ; but one or other of the films al ivays breaks before the 

 little thickish mass which surrounds the plane film and attaches it to the curved 

 films can be entirely absorbed. 



It is here that the experiment stops in the fourth series. Let us now proceed 

 Btill further. Suppose it to be the plane film that breaks ; then the oil of which 

 it is constituted unites itself rapidly with that which formed the rest of the sys- 

 tem, and the whole is reduced to a single film, somewhat thick, which remains 

 attached to the circumference of the two rings, presenting, in the meridian di- 

 rection, a curvature slightly concave, (Fig. 2.) In comparing 

 ""Pig. 2 ggg_ I r^j^jfi 2, it will be observed that in the first the meridian con- 



cavity between the two rings is much more considerable than 

 in the second — a difi'erence easily explained if we consider that 

 the small thick mass which suiTOunds the plane film, and whose 

 meridian section is represented in a, b, c, and a' V c', (Fig. 1,) has three surfaces, 

 which, for the sake of equilibrium, must respectively exert the same pressure. 

 In effect, the curvature in those which have for meridian lines the arcs ab and 

 cb, is evidently concave in all directions around any one point, while the third, 

 whose meridian arc is ac, presents in each point a convex curvature in the 

 direction perpendicular to that arc; whence it is requisite that, in a state of 

 'equilibrium, the concavity of this same arc should be much more decided than 

 that of the arcs ab and cb / but it is clear that this condition could not be ful- 

 fijled with a meridian curvature as weak as that in Fig. 2. 



§ 3. The liquid figure being thus reduced to a single film,* let us gradually 

 raise the upper ring. The film will now be extended, and from being rather 

 thick, as was above said, will become very thin ; the meridian concavity will be 

 more decided, and we shall thus obtain a laminar figure (Fig. 3) whose form re- 

 pj,, 3 calls in every respect the surface to which I have given (4th series, 

 § 14) the name of catcnoid. If we continue to raise the upper 

 ring, we shall an-ive at a point where equilibrium is no longer 

 possible, and we shall see the figure become spontaneously more 

 and more constricted, till at last it separates into two parts which 

 respectively proceed to form a plane film in each of the two rings. In repeating 

 the experiment, I have attempted to arrest the upper ring precisely at the point 

 in question, and have found that then the separation of the two rings was per- 

 ceptibly two-thirds of their diameter ; this ratio, it may be remembered, is that 

 at which I arrived (4th series, § 21 ) as the limitary height of the partial catenoid. 

 § 4. It is a remarkable fact that the disunion of our laminar figure is effected 

 in exactly the same manner with that of full figm'es (2d series, § 62 :) at the mo- 

 ment which precedes this disunion, the constriction is converted into a cylindri- 

 cal thread which is transformed into spherules of different diameters ; only here 

 the thread is itself laminar, as well as the large spherule ; the other spherules 



*For this it is necessary that, in the preceding system, it should be the plane film which is 

 broken ; if it happened that the ruptmo took place in one of the curved films, it would re- 

 main to recommence the experiment, and, in order to operate With certainty, break the plane 

 film with the point of the syringe a little before the spontaneous ruptui'o is expected to take 

 place. 



