WITHDRAWN FROM THE ACTION OP GRAVITY. 217 



of diameter, the film of oil does not break, and the ring then returns upon 

 itself (§ 11) with a much greater rapidity than when the film of oil Is broken, 

 and the ring remains isolated. The traction which the film of oil e:Jserts on the 

 inner circumference of the ring ought therefore to produce an effect analogous 

 to that of the attraction of Saturn, that is to ?ay, contribute to increase the 

 flattening. "Well, the ring of oil before the rupture of the film presents a very- 

 marked flattening. In order to obtain it perfectly, care must be taken that the 

 sphere be well centred in relation to the disc, before beginning the experiment; 

 and it is useful to turn the handle with a velocity somewhat less than that 

 indicated at § 11 ; the most suitable velocity has appeared to me to be about two 

 turns in a second. As soon as the film of oil breaks the flattening disappears, 

 and the generatrix of the ring becomes, as we have seen, sensibly circular.* 



15. Geometricians, Avho have investigated the figiu-e of equilibrium of a liquid 

 mass in rotation, have only regarded ihe case in which the attraction which 

 counteracts the centrifugal force is that of universal gravitation, and they have 

 demonstrated that elliptical figures in that case satisfy this equilibrium. Are 

 we thence to conclude that the annular form developed by the rotation of our 

 mass of oil results from the different law which governs molecular attraction, 

 (§ 10,) and that, in the instance of the heavenly bodies, the figure of an iso- 

 lated ring could not be produced by the sole combination of centrifugal force and 

 of the mutual attractions of the different parts of the mass ? 1 am not of that 

 opdliiou, and I think it, on the contrary, very probable that if calculation could 

 approach the general solution of this great problem, and lead directly to the 

 determhiation of all the possible figures of equilibrium, the annular figure 

 would be included among them. This general and direct solution presenting 

 very great difficulties, geometricians have contented themselves with trying 

 whether elliptical figures could satisfy the equilibrium, and with proving that 

 they in fact do satisfy it ; but they leave the question in doubt, whether other 

 figures would not fulfil the same conditions. In truth, M. Liouville, in his 

 last researches on this subject,! appears at first view to have nearly solved 

 the question, by introducing the consideration of the stability of the figure of 

 equilibrium, and showing that for each value of the moment of rotation, or, in 

 other words, for any initial movement, whatever, of the mass, there is always 

 an elliptical figure, either of revolution or of three unequal axes, according to 



* I bad thought that it would be possible to obtain rings isolated and greatly flattened by 

 operating upon larger masses of oil, for then, the ring having a larger volume,' the inflnence 

 of the molecular attraction should be less. But I have found that, in operating on largei 

 masses, it was necessary, in order to obtain the ring in a regular manner, to employ a more 

 feeble velocity of rotation, so that, if the influence of the molecular attraction was diminished, 

 that of the centrifugal force was so equally. Tlie flattening, then, did not become more sensi- 

 ble ; or, if I have sometimes imagined that I observed any, I have not been able to reproduce 

 it at will. I have operated thus on spheres which were, successively, about 10, 11, 12, and 

 14 U'ntimetrcs in diameter, with discs of a diameter of from seven to nine centimetres, and in 

 a vessel with plane surfaces, having a bottom 35 centimetres square, and a depth of -25 centi- 

 metres. The effects, however, thus obtained f-e very beautiful. The rings are magnificent; 

 present a considerable diameter, and remain suiue;.;:es tor eight to ten seconds before retmn- 

 iug on themselves. With a sphere of ten centimetres diameter, a disc of seven, and a velocity 

 a little less than one turn of the disc per second, we obtain, in a very beautiful and very 

 marked manner, the flattening resulting from the traction of the film of oil. 



These experiments, however, are inconvenient and difficult, on account of the large dimen- 

 sions of the vessel, and the great quantity of alcoholic liquid necessary to till it. 



It may be conceived, moreover, why a larger mass of oil requires a less velocity of rotation 

 to produce a regular ring. It is precisely because the molecular attraction has less influence; 

 whence, it results that, if we attempt to employ tlie same velocity of rotation which would 

 give a beautiful ring with a less quantity of oil, the mass disunites, and is scattered into 

 spherules. 



t The memoir of M. Liouville was communicated to the Academy of Sciences in the sitting 

 of the 13th of February in this year. An analysis of it may be found in the Journal L'/h- 

 stitiU, No. 477. 



