Page:Harper's New Monthly Magazine - v108.djvu/137

is a little shorter than the other. We might also describe the figure as resembling a short and fat cigar, with rounded ends, in rotation about a pin which transfixes it through the middle.

As we spin the body yet slower, the egg or cigar gets longer and thinner, and it continues to be stable. But Poincaré showed that the stability will cease when the elongation has proceeded to an extent which he left undetermined. It appears from my own recent investigations that when the stability ceases the length of the cigar or egg is about three times its breadth. This is illustrated by the dotted lines in Fig. 3; the lower figure shows the section of the egg passing through the axis of rotation, and the upper one that at right angles to the axis. Poincaré's principle now again becomes applicable, and we know that there must here occur a new series of figures coalescent with the egglike forms. His conjectural sketch of the new figure resembled a pear, and although in the accurate drawing, shown by the firm lines in Fig. 3, the resemblance to a pear is not very great, it is convenient to call it the pear shape. In describing the figure as pear-shaped I mean that one end of the egglike form from which it emanates has become a little more pointed, and the other end has become blunted; also, the figure is slightly depressed on the stalk side of the middle, and slightly swollen on the other side. It might be supposed that we could at once assert that "exchange of stability" has taken place and that these pear-shaped figures are stable. But there are certain cases in which further consideration is needed to discover whether or not this occurs, and the present is an instance of these dubious cases. It was indeed prodigiously laborious to carry out the task, but at length I succeeded in proving the pear shape to be stable.

It will probably seem to some persons an extraordinary waste of time that a man should be willing to spend two years, as I have done, in endeavoring to determine a possible form of an ideal mass of liquid rotating in space. The field of research is apparently narrow and the labor was great, yet it may be maintained that it is only in some such way as this that we shall ever be able to understand the processes of celestial evolution.

We now know that stability is shunted off into the pear series after leaving the cigar or egg series. As the rotation slackens still further, the stalk of the pear tends to become more prolonged, whilst the changes elsewhere are inconspicuous.

Fig. 2.—Diagram illustrative of the meaning
of Coalescence

No one has as yet succeeded in tracing the developments of these figures further, and the incidence of a new epoch of instability is therefore unknown. There is, however, strong reason to suppose that a piece of the stalk will finally detach itself and will form a satellite. This will become more intelligible when I speak of another investigation, but the chain of reasoning has been somewhat long, and it has involved conceptions which must be unfamiliar to most readers; I therefore think it well to illustrate the argument diagrammatically in Fig. 4. This figure does not in any sense represent the planetary, ellipsoidal, and pear-shaped figures themselves, and is not drawn to any definite scale, but is intended merely to show the sequence of ideas. We began with a sphere of liquid at rest and then caused it to rotate, gradually increasing the speed of spinning. This is illustrated by the horizontal line from left to right. The form of the liquid mass is planetary, with gradually increasing oblateness. At the point marked with the name of Jacobi stability just ceases, and the dotted continuation of the straight line indicates that the flatter planetary bodies are unstable. The mathematician Jacobi discovered this point of coalescence, although he certainly did not investigate the stability of the coalescent series.