FIGURE AND STABILITY OF A LIQUID SATELLITE. 
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problem .'* He finds solutions perfectly analogous to Maclaurin s and Jacobis 
ellipsoids and to the pear-shaped figure. In consequence of the greater simplicity of 
the conditions, he is able to follow the development of the cylinder of pear-shaped 
section until the neck joining the two parts has become quite thin. His analysis, 
besides, points to the rupture of the neck, although the method fails to afford the 
actual shapes and dimensions in this last stage of development. 
He is aide to prove conclusively that the cylinder of pear-shaped section is stable, 
and it is important in connection with our present investigation to note that he finds 
no evidence of any break in the stability of that cylinder up to its division into two 
parts. 
The stability of Maclaurin’s and of the shorter Jacobian ellipsoids is, of course, 
well established, and I imagined that the pear-shaped figure with incipient furrowing 
was also proved to lie stable. But M. Liapounoff now states! that he is able to 
prove the pear-shaped figure to be unstable from the beginning, and lie attributes the 
discrepancy between our conclusions to the fact that my result depended on the 
supposed rapid convergency of an infinite series, of which only a few terms were 
computed. The terms computed diminish rapidly, and it seemed to me evident that 
the rapid diminution must continue, so that I feel unable to accept the hypothesis 
that the sum of the neglected terms could possibly amount to the very considerable 
total which would be necessary to reverse my conclusion. I am, therefore, still of 
opinion that the pear-shaped figure is stable at the beginning; and this view receives 
a powerful confirmation from Mr. Jeans’s researches. The final decision must await 
the publication of M. Liapounoff’s investigation. 
But there is another difficulty raised by the present paper. I had fully expected to 
find an approximation to a stable figure consisting of two bulbs joined by a thin neck, 
but while my work indicates the existence of such a figure, it seems to me, at present, 
conclusive against its stability. The weightless pipe joining two bulbs of fluid is 
clearly only a crude representative of a neck of fluid, but I find it hard to imagine 
that it is so very imperfect that the reality should lie stable, while the representation 
is unstable. My present investigation shows that two quasi-ellipsoids just detached 
from one another do not possess secular stability. The vertices of such bodies 
would be blunt points nearly in contact; the introduction of a short pipe without 
weight between these blunt points would differ exceedingly little from two sharp 
points actually in contact. Is it possible that the difference would produce all the 
change from great instability even to limiting stability ? The opening of a channel 
between the two masses is the removal of a constraint; the system does not possess 
true secular stability when the channel is closed, and we should have to believe that 
the removal of a constraint induces stability ; and this is, I think, impossible. 
If, then, Mr. Jeans is right in believing in the stable transition from the single 
* “On the Equilibrium of Rotating Liquid Cylinders,” ‘Phil. Trans.,’ A, vol. 200, pp. 67-104. 
t See reference in Preface. 
