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Further illustrations of this subject might be given； but enough has been said to explain the nature of the conclusions which have been drawn from this class of observation。
In my account of these remarkable systems the consideration of one very important conclusion has been purposely deferred。 Since the light…curve is explicable by eclipses； it follows that the sizes of the two stars are determinable relatively to the distance between them。 The period of their orbital motion is known； being identical with the complete period of the variability of their light； and an easy application of Kepler's law of periodic times enables us to compute the sum of the masses of the two stars divided by the cube of the distance between their centres。 Now the sizes of the bodies being known； the mean density of the whole system may be calculated。 In every case that density has been found to be much less than the sun's； and indeed the average of a number of mean densities which have been determined only amounts to one…eighth of that of the sun。 In some cases the density is extremely small； and in no case is it quite so great as half the solar density。
It would be absurd to suppose that these stars can be uniform in density throughout； and from all that is known of celestial bodies it is probable that they are gaseous in their external parts with great condensation towards their centres。 This conclusion is confirmed by arguments drawn from the theory of rotating masses of liquid。 （See J。H。 Jeans； 〃On the density of Algol variables〃； 〃Astrophysical Journ。〃 Vol。 XXII。 （1905）； page 97。）
Although； as already explained； a good deal is known about the shapes and the stability of figures consisting of homogeneous incompressible liquid in rotation； yet comparatively little has hitherto been discovered about the equilibrium of rotating gaseous stars。 The figures calculated for homogeneous liquid can obviously only be taken to afford a general indication of the kind of figure which we might expect to find in the stellar universe。 Thus the dotted curve in Fig。 5； which exhibits one of the figures which I calculated； has some interest when placed alongside the figures of the stars in RR Centauri； as computed from the observations； but it must not be accepted as the calculated form of such a system。 I have moreover proved more recently that such a figure of homogeneous liquid is unstable。 Notwithstanding this instability it does not necessarily follow that the analogous figure for compressible fluid is also unstable； as will be pointed out more fully hereafter。
Professor Jeans has discussed in a paper of great ability the difficult problems offered by the conditions of equilibrium and of stability of a spherical nebula。 （〃Phil。 Trans。 R。S。〃 Vol。 CXCIX。 A （1902）； page 1。 See also A。 Roberts； 〃S。 African Assoc。 Adv。 Sci。〃 Vol。 I。 （1903）； page 6。） In a later paper （〃Astrophysical Journ。〃 Vol。 XXII。 （1905）； page 97。）； in contrasting the conditions which must govern the fission of a star into two parts when the star is gaseous and compressible with the corresponding conditions in the case of incompressible liquid； he points out that for a gaseous star （the agency which effects the separation will no longer be rotation alone； gravitation also will tend towards separation。。。From numerical results obtained in the various papers of my own；。。。I have been led to the conclusion that a gravitational instability of the kind described must be regarded as the primary agent at work in the actual evolution of the universe； Laplace's rotation playing only the secondary part of separating the primary and satellite after the birth of the satellite。〃
It is desirable to add a word in explanation of the expression 〃gravitational instability〃 in this passage。 It means that when the concentration of a gaseous nebula （without rotation） has proceeded to a certain stage； the arrangement in spherical layers of equal density becomes unstable； and a form of bifurcation has been reached。 For further concentration concentric spherical layers become unstable； and the new stable form involves a concentration about two centres。 The first sign of this change is that the spherical layers cease to be quite concentric and then the layers of equal density begin to assume a somewhat pear…shaped form analogous to that which we found to occur under rotation for an incompressible liquid。 Accordingly it appears that while a sphere of liquid is stable a sphere of gas may become unstable。 Thus the conditions of stability are different in these two simple cases； and it is likely that while certain forms of rotating liquid are unstable the analogous forms for gas may be stable。 This furnishes a reason why it is worth while to consider the unstable forms of rotating liquid。
There can I think be little doubt but that Jeans is right in looking to gravitational instability as the primary cause of fission； but when we consider that a binary system； with a mass larger than the sun's； is found to rotate in a few hours； there seems reason to look to rotation as a contributory cause scarcely less important than the primary one。
With the present extent of our knowledge it is only possible to reconstruct the processes of the evolution of stars by means of inferences drawn from several sources。 We have first to rely on the general principles of stability； according to which we are to look for a series of families of forms； each terminating in an unstable form； which itself becomes the starting…point of the next family of stable forms。 Secondly we have as a guide the analogy of the successive changes in the evolution of ideal liquid stars； and thirdly we already possess some slender knowledge as to the equilibrium of gaseous stars。
From these data it is possible to build up in outline the probable history of binary stars。 Originally the star must have been single； it must have been widely diffused； and must have been endowed with a slow rotation。 In this condition the strata of equal density must have been of the planetary form。 As it cooled and contracted the symmetry round the axis of rotation must have become unstable； through the effects of gravitation； assisted perhaps by the increasing speed of rotation。 （I learn from Professor Jeans that he now （December 1908） believes that he can prove that some small amount of rotation is necessary to induce instability in the symmetrical arrangement。） The strata of equal density must then become somewhat pear… shaped； and afterwards like an hour…glass； with the constriction more pronounced in the internal than in the external strata。 The constrictions of the successive strata then begin to rupture from the inside progressively outwards； and when at length all are ruptured we have the twin stars portrayed by Roberts and by others。
As we have seen； the study of the forms of equilibrium of rotating liquid is almost complete； and Jeans has made a good beginning in the investigation of the equilibrium of gaseous stars； but much more remains to be discovered。 The field for the mathematician is a wide one； and in proportion as the very arduous exploration of that field is attained so will our knowledge of the processes of cosmical