STELLAR EVOLUTION JEANS. 161 



comparable with that of iron, it passes through a critical point at 

 which there is a sudden swing over from one type of break-up to 

 the other. This critical point occurs when the density of the star 

 has become such that the ordinary gas laws are substantially de- 

 parted from throughout the greater part of the star's interior. This 

 density is, however, precisely that which marks the demarcation 

 between giant and dwarf stars. Thus the general conclusion of 

 abstract theory is that a giant star will break up under increasing 

 rotation in the way we have already had under consideration, but 

 that a dwarf star will break up in the same way as a homogeneous 

 incompressible mass, such as a mass of water. 



The discovery of the method of break-up in this second case forms 

 one of the most difficult problems of applied mathematics. In spite 

 of the labors of man}^ eminent mathematicians, among whom may be 

 mentioned Maclaurin, Jacobi, Kelvin, Poincare, and G. H. Darwin, 

 the problem is still far from complete solution. It is found that, 

 as the rotation of a homogeneous mass increases, the boundary re- 

 mains of exact spheroidal shape until an eccentricity of 0.8127 is 

 reached, at which the axes are in the ratio of about 12 : 12 : 7. With 

 a further increase of rotation the boundary ceases to be a figure of 

 revolution; it becomes ellipsoidal and retains an exact ellipsoidal 

 shape until the axes are in a ratio of about 23 : 10 : 8. Beyond this 

 it is impossible for the mass to rotate in relative equilibrium at all, 

 and dynamical motion of some kind must ensue. At first a furrow 

 forms round the ellipsoid in a cross-section perpendicular to the 

 longest axis, but the cross-section in which the furrow appears does 

 not divide the figure symmetrically into equal halves. The furrow 

 deepens, and at this stage the problem eludes exact mathematical 

 treatment. It appears highly probable, although it cannot be rigor- 

 ously proved, that the furrow will continue to deepen until it sepa- 

 rates the figure into two unequal masses. On the assumption that 

 this is what would actually happen we may conjecture that the 

 process we have been describing is that of the fission of a single 

 star into a binary of the familiar type, but the conjecture is beset by 

 many difficulties. To mention one only : if we have truly described 

 the history of a star before fission, the star ought during a mod- 

 erate part of its life to possess an ellipsoidal figure, and as this 

 rotated the light received from the star ought to vary to an extent 

 which just before fission might amount to 0.9 magnitude. Yet I 

 believe there are only three known stars whose variation of light is 

 such as could possibly be accounted for by an ellipsoidal surface, 

 and even in these cases the interpretation is doubtful. On the other 

 hand, very considerable reassurance is provided by the researches 

 of Russell on multiple stars. After a star has broken into two 



