Page:Harper's New Monthly Magazine - v108.djvu/138

From this point the line of stability is drawn backwards so as to indicate that we are to retard the rotation instead of further accelerating it. We now pursue the stable egglike series of Jacobi to a new junction of vanishing stability. This I have marked with the name of Poincaré and with my own name because, while he had the far greater merit of indicating that such a junction must exist and of pointing out its nature, I was the first to reduce his ideas to numbers. From this point the more elongated Jacobian figures become unstable, as indicated by the dotted line, whilst the stability is shunted off into the series of pear-shaped figures.

The next point of junction is as yet undetermined, but there is good reason to suppose that the pear-shaped body separates into two, and that the smaller will revolve round the larger as a satellite.

I now turn to another investigation bearing on this subject.

It occurred to Mr. Jeans, a young Cambridge mathematician, that the solution of a relatively simple problem would throw much light on the later transformations of the pear-shaped figures. His problem involves conditions which are much more ideal than those of the comparatively realistic problem which we have been considering, but when I have explained his problem I think it will be admitted that his expectations have been fully realized.

Fig. 3—Sections of the Pear-shaped Figure

An absolutely infinite cylinder of water at rest would maintain its shape under its own gravitation if its section were circular, and if it were very remote from all other bodies throughout its whole length. Such an infinite cylindrical column is stable as regards the circularity of its section, so that if throughout its whole length it were deformed in any and the same way it would perform oscillations about the circular shape. But it is not stable as to the straightness of its axis, and throughout the changes which we are going to follow I suppose that there is some supernatural being who restrains the tendency of the column to bend, and thus maintains the form as perfectly columnar. The infinite straight circular cylinder of water is analogous to the perfect sphere of water, and both are stable in shape, the first as to the circularity of its section, the second as to the sphericity of its form. Mr. Jeans now imagines the infinite cylinder to be set in rotation about its infinitely long axis, and he finds that so long as the rotation is less rapid than a certain speed the circularity of section and its stability are maintained. These circular and stable rotating cylinders are strictly analogous with the planetary rotating bodies we were considering previously.

When the rotation has increased to a certain amount the stability ceases, and he finds a new series of cylinders which are elliptic in section. At first the ellipticity or flattening is infinitely small, so that at the stage where the stability ceases the circular and elliptic cylinders are coalescent. These elliptic cylinders are stable, and are perfectly analogous with the egg-shaped figures of Jacobi.

To pursue these figures, we must cause the speed of rotation to slacken, just as we did before, and we then find the ellipticity of section of the infinite cylinder increasing, exactly as the figures of Jacobi became more elongated as the speed of rotation diminished. When the ro-