type, whilst that of the moon should be regarded as more nearly resembling the second type.
Accordingly, in the present article I propose to show what light is thrown on the second of these supposed types of genesis by certain recent mathematical investigations.
The results which will be stated are certain and absolute, but the degree of their applicability to celestial history is necessarily a matter of speculation on which it is impossible to pronounce an unhesitating opinion. It is, however, clear that while mere general impression and conjecture as to mechanical possibilities would have but little weight, yet speculation will assume a value of an entirely different order when it is founded on absolute certainty as to the mechanical properties of a celestial body of even ideal simplicity. It thus becomes worth while to expend a great deal of thought and labor upon the discussion of the mechanics of such a system. I here, then, take leave of the heavens and betake myself to concrete and simple cases of matter in motion.
Let us imagine that there is in space a mass of liquid such as water; let it be very far removed from all other bodies, and let it be rotating, all in one piece, as though it were frozen. The problem to be considered is the determination of the shape it will assume under the influence of its own gravitation and of rotation. We have further to discover whether any shape which it may be capable of assuming is stable. It is said to be stable if it trembles like a jelly after disturbance but still maintains the original form. It is unstable if, after disturbance, it completely changes its shape and assumes some wholly different one. If a certain shape of rotating liquid were found to be possible, it could not be regarded as typifying one of the stages of celestial evolution unless it were also a shape which could continue to subsist when subjected to any small disturbance.
The problem, then, is to determine the stable forms of rotating liquid; but it will appear below that this cannot be done without reference to other unstable forms. In fact, the problem needs systematic solution, and it will not suffice to pick out for discussion one small portion of the whole.
It may seem rather illogical to say that the simplest case of rotating liquid is when the liquid is at rest; but such a statement would be correct, for we may imagine the liquid to rotate slower and slower and finally to stop, and there will be no violent contrast between the case of very slow rotation and none at all. We begin, then, with liquid at rest. The mutual gravitation of the water is the only force which acts on the system. The water will obviously crowd together into the smallest possible space, so that every particle may get as near the centre as its neighbors will let it. I suppose the water to be incompressible, so that the central portion, although under pressure, does not become more dense. Since there is no upward or downward or right or left about the system, it must be symmetrical in every direction, and the only figure which possesses this property of universal symmetry is the sphere. Further, if the sphere were slightly deformed and then released from constraint it would oscillate to and fro, so that on the average the figure would remain spherical. After a time the friction of the liquid would of course annul the oscillations. A sphere is, then, said to be a stable figure of equilibrium of a mass of liquid at rest. Before going further it will be well to explain that the size and density of the liquid celestial bodies under consideration are immaterial, for the theory of the forms assumed is unaffected by these considerations.
Now if the sphere of liquid be made to rotate it will become slightly flattened like an orange, or like the earth itself. This flattening takes place because the rotation tends to make the equator fly outwards—a tendency which is restrained by gravity. Such a planetary figure is stable, and, in fact, it typifies the figures of actual planets.
If we quicken the rotation of the planetary figure, the degree of the flattening increases, and the stability continues to subsist for a time, but the tendency to spring back to the primitive figure after disturbance is weakened. At length we come to such a degree of flattening that all the spring is gone, and the slightest disturbance will
