218 THE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 



the circumstauces, which constitutes a form of stable equilibrium. It appears, 

 in C'flect, uatural to admit that, for a given disturbance of a liquid mass, there is 

 but one single final state admissible ; and, in this case, this state must necessa- 

 rily possess stability. However, I do not deem the conclusion which may be 

 drawn from these results so general as it appears at first sight. Without doubt, 

 for a primitive disturbance given, there is only one final stale possible, and that 

 state must be stable, lint the condition of stability of a found figure of equi- 

 librium docs not necessarily involve the consequence that this figure will con 

 stitute the final state in quet?tion, for it may happen that several figures of equi- 

 librium, corresponding to the same primitive disturbance, might equally possess 

 stability, and that the choice of the mass for one of these figures may have been 

 determined by other circumstances ; for example, by the modifications which 

 its movement experiences in the first moments of rotation. In fact, it is by 

 examining these modifications, to which the attention of geometricians has not 

 been directed, that I shall attempt to arrive at the mode of generation of annular 

 figures. 



16. When the mass begins to revolve upon itself, the angular velocity of the 

 portions remote from the axis, which are carried off by their centrifugal force, 

 necessarily goes on diminishing. This diminution is especially apparent ou 

 the equator of the mass, and it is the more considerable in proportion as the 

 initial movement of rotation was more rapid. It thence results that, in the first 

 instants of a sufliciently rapid rotation, there will be a great diilerenco of angular 

 velocity between the portions which are near the axis and those Avhich are near 

 the equator. Nevertheless, if we admit for a moment that, in virtue of the 

 adherence of the liquid for itself, and of the friction of its several parts, the 

 portions which turn most rapidly communicate by degrees a part of their ve- 

 locity to the others, so that in the end the result is a mean angular velocity, 

 corresponding to the same moment of rotation, and equal in all the points of the 

 mass, this may take an ellipsoidal figure. But long before the feeble forces, of 

 whicb wc have just spoken, can bring about this mean result, another order of 

 phenomena would be manifested, which may impede the development of the 

 elliptical figure and give rise to an annular form. 



In fact, it follows necessarily from the preceding considerations that, in the 

 first instants of a rotation aufticicntly rapid, the centrifugal force at the equator 

 of the mass will be much less than that which would correspond to the above 

 mean velocity ; and that, on the other hand, the centrifugal force of the portions 

 near the axis will be by much superior to that which would correspond to the 

 same mean velocity. The liquid next the axis will, therefore, be driven towards 

 the liquid of the equator, whence there will necessarily result the formation of 

 a sort of circular cushion, (bourrdei,) more or less marked. In other words, 

 the mass will soon become hollow in the middle, and will swell out all around. 

 Now as soon as this phenomenon takes place, it will be conceived that the at- 

 traction exerted by this hourrclct on the liquid remaining around the axis must 

 be an addition to the action of the centrifugal force, and contribute to increase 

 the volume of the hourrtlet at the expense of the central, liquid. Hence, there- 

 fore, it may evidently result that all the liquid will leave the axis for the hour- 

 rclct, and the latter become in a manner a veritable ring. 



This generation of the annular figures would therefore be independent of 

 the law which the attraction follows, and would be, in consequence, the same 

 in the case of universal attraction and in that of molecular attraction. 



17. It is easy to verify this mode of generation upon our mass of oil, or at 

 least to assure ourselves that during the formation of the hourrclct and of the 

 ring, the angular velocity is much less at the equator of the mass than toAvards 

 the axis. For this purpose I shall first point out that Avhcn a certain number 

 of experiments have been performed upon the same mass of oil, and this has 

 been several times disunited and refoimed into a single sphere and into a ring. 



