30 PLATEAU ON THE FIGURE OF A LIQUID MASS 
15. Geometricians who have investigated the figure of equi- 
librium of a liquid mass in rotation, have only regarded the 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 rota- 
tion of our mass of oil results from the different law which governs 
molecular attraction (§ 10), and that, in the instance of the hea- 
venly bodies, the figure of an isolated ring could not be pro- 
duced by the sole combination of centrifugal force and of the 
mutual attractions of the different parts of the mass? I am not 
of that opinion, 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 determination of all the 
possible figures of equilibrium, the annular figure would be in- 
cluded among them. This general and direct solution presenting 
very great difficulties, geometricians have contented themselves 
with trying whether elliptical figures could satisfy the equili- 
brium, and with proving that they in fact do satisfy it; but 
they leave the question in doubt, whether other figures would 
not fulfill 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 the circumstances, which constitutes a form of 
stable equilibrium. It appears, in effect, natural 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 state possible, and that state must be 
stable; but the condition of stability of a found figure of equi- 
librium does not necessarily involve the consequence that this 
ploy the same velocity of rotation which would give a beautiful ring with a less 
quantity of oil, the mass disunites, and is scattered into spherules. 
* The memoir of M. Liouville was communicated to the Academy of Sciences 
in the sitttng of the 18th of February in this year. An analysis of it may be 
found in the Journal L’ Institut, No. 477. 
