Page:Harper's New Monthly Magazine - v108.djvu/136

become greater and greater, so that the figure will break up in some way which cannot be determined. Fig. 1 shows the outline of the planetary figure when it has reached the critical stage and has just become unstable. From the sphere to this critical planetary figure out course is clear and we have found the stable forms, but we have now to consider the further stages of development.

The reader will have noticed that the figures occur in a continuous series, and gradually undergo a change as the rotation increases. Now Poincaré, the great French mathematician, has proved a principle of very wide generality, of which the application to the particular instance in hand is as follows: if a series of figures be followed, and if it is found that at a certain stage of development there is a change from stability to instability, then we have notice that there is another sort of figure coalescent with the first sort at the particular moment of change. He also proved a still more important point, namely, that in general the stability which deserts the first series of figures passes on into the new series. The meaning of the term "coalescent" in this statement may perhaps be somewhat difficult of apprehension, and it seems advisable to explain it more exactly. The ellipses drawn along the horizontal line in Fig. 2 form a series. The middle one of these figures as we go from left to right is the one where the ellipses cease to be flattened vertically and begin to be flattened horizontally. It is, in fact, a circle, although it occurs in a series of ellipses.

Fig. 1.Limiting stable Planetary Figure

Now consider the series of circles drawn one above another. I have thought it sufficient to draw only three of them, as that number suffices to indicate that they belong to a series which shrinks in size as we go downwards. The middle circle of this series is identical with the circle which belongs to the series of ellipses, and the two series of figures, which are essentially independent of one another, nevertheless coalesce at the middle circle. This illustration has nothing to do with our problem, except as affording an explanation of the meaning of coalescence.

Let us now return to the consideration of the rotating liquid: When the stability deserts the planetary figure the new sort of figure is found by elongating one of the equatorial diameters of the planet and shortening that at right angles to it. Thus the section of the equator is no longer a circle, but is an ellipse. The new figure is an ellipsoid with three unequal axes, but in its initial condition when it is coalescent with the planetary figure the two equatorial axes differ infinitely little in length. Thus the planetary figure and this new figure are coalescent at this stage.

According to Poincaré's principle of "exchange of stabilities," the stability which has deserted the planetary figure has passed over to the new set of figures, which have three unequal axes. It is remarkable that the speed of rotation in this new set of figures is less than it was, so that in order to follow the stable series of figures for varied rotation it is necessary to increase the rate of spinning up to the stage when the planetary figure becomes unstable, and then to diminish it.

Now let us follow the stable ellipsoidal figures. As we gradually slacken the speed of rotation the former equator of the planetary figure becomes more and more flattened along one diameter and elongated along another at right angles to it, so that the shape becomes like that of an egg (but with both ends alike), and it rotates about an axis through the shortest diameter of the egg. Speaking more exactly, the circumference of the egg must not be supposed to be exactly round, but that diameter of its girth which coincides with the axis of rotation