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Stars and planets are formed of materials which yield to the enormous forces called into play by gravity and rotation。 This is obviously true if they are gaseous or fluid； and even solid matter becomes plastic under sufficiently great stresses。 Nothing approaching a complete study of the equilibrium of a heterogeneous star has yet been found possible； and we are driven to consider only bodies of simpler construction。 I shall begin therefore by explaining what is known about the shapes which may be assumed by a mass of incompressible liquid of uniform density under the influences of gravity and of rotation。 Such a liquid mass may be regarded as an ideal star； which resembles a real star in the fact that it is formed of gravitating and rotating matter； and because its shape results from the forces to which it is subject。 It is unlike a star in that it possesses the attributes of incompressibility and of uniform density。 The difference between the real and the ideal is doubtless great； yet the similarity is great enough to allow us to extend many of the conclusions as to ideal liquid stars to the conditions which must hold good in reality。 Thus with the object of obtaining some insight into actuality； it is justifiable to discuss an avowedly ideal problem at some length。
The attraction of gravity alone tends to make a mass of liquid assume the shape of a sphere； and the effects of rotation； summarised under the name of centrifugal force； are such that the liquid seeks to spread itself outwards from the axis of rotation。 It is a singular fact that it is unnecessary to take any account of the size of the mass of liquid under consideration； because the shape assumed is exactly the same whether the mass be small or large； and this renders the statement of results much easier than would otherwise be the case。
A mass of liquid at rest will obviously assume the shape of a sphere； under the influence of gravitation； and it is a stable form； because any oscillation of the liquid which might be started would gradually die away under the influence of friction； however small。 If now we impart to the whole mass of liquid a small speed of rotation about some axis； which may be called the polar axis； in such a way that there are no internal currents and so that it spins in the same way as if it were solid； the shape will become slightly flattened like an orange。 Although the earth and the other planets are not homogeneous they behave in the same way； and are flattened at the poles and protuberant at the equator。 This shape may therefore conveniently be described as planetary。
If the planetary body be slightly deformed the forces of restitution are slightly less than they were for the sphere； the shape is stable but somewhat less so than the sphere。 We have then a planetary spheroid； rotating slowly； slightly flattened at the poles； with a high degree of stability； and possessing a certain amount of rotational momentum。 Let us suppose this ideal liquid star to be somewhere in stellar space far removed from all other bodies； then it is subject to no external forces； and any change which ensues must come from inside。 Now the amount of rotational momentum existing in a system in motion can neither be created nor destroyed by any internal causes； and therefore； whatever happens； the amount of rotational momentum possessed by the star must remain absolutely constant。
A real star radiates heat； and as it cools it shrinks。 Let us suppose then that our ideal star also radiates and shrinks； but let the process proceed so slowly that any internal currents generated in the liquid by the cooling are annulled so quickly by fluid friction as to be insignificant； further let the liquid always remain at any instant incompressible and homogeneous。 All that we are concerned with is that； as time passes； the liquid star shrinks； rotates in one piece as if it were solid； and remains incompressible and homogeneous。 The condition is of course artificial； but it represents the actual processes of nature as well as may be； consistently with the postulated incompressibility and homogeneity。 （Mathematicians are accustomed to regard the density as constant and the rotational momentum as increasing。 But the way of looking at the matter； which I have adopted； is easier of comprehension； and it comes to the same in the end。）
The shrinkage of a constant mass of matter involves an increase of its density； and we have therefore to trace the changes which supervene as the star shrinks； and as the liquid of which it is composed increases in density。 The shrinkage will； in ordinary parlance； bring the weights nearer to the axis of rotation。 Hence in order to keep up the rotational momentum； which as we have seen must remain constant； the mass must rotate quicker。 The greater speed of rotation augments the importance of centrifugal force compared with that of gravity； and as the flattening of the planetary spheroid was due to centrifugal force； that flattening is increased； in other words the ellipticity of the planetary spheroid increases。
As the shrinkage and corresponding increase of density proceed； the planetary spheroid becomes more and more elliptic； and the succession of forms constitutes a family classified according to the density of the liquid。 The specific mark of this family is the flattening or ellipticity。
Now consider the stability of the system； we have seen that the spheroid with a slow rotation； which forms our starting…point； was slightly less stable than the sphere； and as we proceed through the family of ever flatter ellipsoids the stability continues to diminish。 At length when it has assumed the shape shown in a figure titled 〃Planetary spheroid just becoming unstable〃 （Fig。 2。） where the equatorial and polar axes are proportional to the numbers 1000 and 583； the stability has just disappeared。 According to the general principle explained above this is a form of bifurcation； and corresponds to the form denoted A。 The specific difference a of this family must be regarded as the excess of the ellipticity of this figure above that of all the earlier ones； beginning with the slightly flattened planetary spheroid。 Accordingly the specific difference a of the family has gradually diminished from the beginning and vanishes at this stage。
According to Poincare's principle the vanishing of the stability serves us with notice that we have reached a figure of bifurcation； and it becomes necessary to inquire what is the nature of the specific difference of the new family of figures which must be coalescent with the old one at this stage。 This difference is found to reside in the fact that the equator； which in the planetary family has hitherto been circular in section； tends to become elliptic。 Hitherto the rotational momentum has been kept up to its constant value partly by greater speed of rotation and partly by a symmetrical bulging of the equator。 But now while the speed of rotation still increases （The mathematician familiar with Jacobi's ellipsoid will find that this is correct； although in the usual mode of exposition； alluded to above in a footnote； the speed diminishes。）； the equator tends to bulge outwards at two diametrically opposite points and to be flattened midway between these protuberances。 The specific difference in the new family； denoted in the general sketch by b； is this ellipticity of the equator。 If we had traced the planetary figures with circular equators beyond this stage A； we should have found them to have become unstable； and the stability has been shunted off along the A ＋ b family of forms with elliptic equators。
This new series of figures； generally named after the great mathematician Jacobi； is at first only just stable； but as the density increases the stability increases； reaches a maximum and then declines。 As this goes on the equator of these Jacobian figures becomes more and more elliptic； so that the shape is considerably elongated in a direction at right angles to the axis of rotation。
At length when the longest axis of the three has become about three times as long as the shortest （The three axes of the ellipsoid are then proportional to 1000； 432； 343。）； the stability of this family of figures vanishes； and we have reached a new form of bifurcation and must look for a new type of figure along which the stable development will presumably extend。 Two sections of this critical Jacobian figure； which is a figure of bifurcation； are shown by the dotted lines in a figure titled 〃The 'pear…shaped figure' and the Jocobian figure from which it is derived〃 （Fig。 3。） comprising two figures； one above the other： the upper figure is the equatorial section at right angles to the axis of rotation； the lower figure is a section through the axis。
Now Poincare has proved that the new type of figure is to be derived from the figure of bifurcation by causing one of the ends to be prolonged into a snout and by bluntening the other end。 The snout forms a sort of stalk； and between the stalk and the axis of rotation the surface is somewhat flattened。 These are the characteristics of a pear； and the figure has therefore been called the 〃pear…shaped figure of equilibrium。〃 The firm line shows