of a Liquid Mass without Weight. 361 



I have verified a consequence of M. Bonnet's researches — 

 namely, that an infinity of surfaces having the mean curvature 

 zero can always pass through a closed outline, either plane or 

 not plane, of absolutely any form. I have had the strangest and 

 most complicated closed outlines made of iron wire, and on 

 issuing from the glycerine-solution each of them was found to 

 be occupied by a single film : this experiment proves, in the first 

 place, that, given any closed outline whatever, there is always at 

 least one surface of mean curvature zero a finite portion of which 

 can fill it. I next show how to make the film undergo as many 

 changes of shape as may be wished without its equilibrium being 

 destroyed, and without its ceasing to rest upon the whole of the 

 closed outline ; but I show that it is no longer a finite portion of 

 each of these new surfaces that occupies the given outline. 



I had found by calculation that the volume of the limiting ca- 

 tenoid is half that of the cylinder on the same base and of the 

 same height ; and I have verified this result by means of a full 

 catenoid of oil, formed between two disks within the alcoholic 

 solution, and having the limiting height ; the disks were then 

 brought nearer until the liquid mass formed a cylinder ; and the 

 height of this cylinder was found to be half that of the catenoid. 



Lastly, by the use of suitable solid frameworks, I have realized 

 a portion of one of M. Lamarle's helicoids, likewise by means of 

 oil surrounded by the alcoholic solution. 



Eleventh Series. — Limits of stability of figures of equilibrium. 

 — General theory of the stability of these figures, — Stability of 

 systems of films. — Stability in cases when gravity comes into play. 



As might be seen from the preceding Series, the sphere is very 

 probably the only closed figure of equilibrium, all the rest having 

 infinite dimensions in certain directions. When the attempt is 

 made to realize partially one of these last, either by means of oil 

 surrounded by the alcoholic solution, or with a film of the gly- 

 cerine-solution in air, it is generally found that, if the solid ter- 

 minations to which the mass or the film adheres are placed so 

 as to comprise too great a portion of the figure, the latter will 

 not form ; whence we must conclude that, with the terminations 

 separated to this extent, it would be unstable. In the present 

 Series I investigate, in the first place, by aid of experiment, cal- 

 culation, and reasoning, the limits of stability of most of the 

 figures of equilibrium that I have studied, and especially of the 

 figures of revolution contained between two equal bases perpen- 

 dicular to the axis. 



When a sphere of oil is freely suspended in the alcoholic mix- 

 ture, it always exhibits perfect stability of form. If this form is 

 altered bymovements imparted to the surrounding liquid, themass 



Phil. Mag, S. 4. Vol. 40. No. 268. Nov. 1870. 2 B 



