The pear-shaped figure 553 
but I do not know at what stage of its development it becomes 
unstable. 
Professor Jeans has solved a problem which is of interest 
as throwing light on the future development of the pear-shaped 
figure, although it is of a still more ideal character than the one 
which has been discussed. He imagines an infinitely long circular 
cylinder of liquid to be in rotation about its central axis. The 
existence is virtually postulated of a demon who is always occupied 
in keeping the axis of the cylinder straight, so that Jeans has only 
to concern himself with the stability of the form of the section of 
the cylinder, which as I have said is a circle with the axis of rotation 
at the centre. He then supposes the liquid forming the cylinder to 
shrink in diameter, just as we have done, and finds that the speed of 
rotation must increase so as to keep up the constancy of the rotational 
momentum. The circularity of section is at first stable, but as the 
shrinkage proceeds the stability diminishes and at length vanishes. 
This stage in the process is a form of bifurcation, and the stability 
passes over to a new series consisting of cylinders which are 
elliptic in section. The circular cylinders are exactly analogous with 
our planetary spheroids, and the elliptic ones with the Jacobian 
ellipsoids. 
Fig. 4. 
Section of a rotating cylinder of liquid. 
With further shrinkage the elliptic cylinders become unstable, 
a new form of bifurcation is reached, and the stability passes over 
to a series of cylinders whose section is pear-shaped. Thus far the 
analogy is complete between our problem and Jeans’s, and in con- 
sequence of the greater simplicity of the conditions, he is able to 
carry his investigation further. He finds that the stalk end of the 
pear-like section continues to protrude more and more, and the 
flattening between it and the axis of rotation becomes a constriction. 
Finally the neck breaks and a satellite cylinder is born. Jeans’s 
figure for an advanced stage of development is shown in Fig. 4, but 
exist with stability for greater rotational momentum. My own work seems to indicate 
that the opposite is true, and, notwithstanding M. Liapounoff’s deservedly great authority, 
I venture to state the conclusions in accordance with my own work. 
