FIGURES FOR HETEROGENEOUS FLUIDS. 147 



answer to this question, but in view of the fact that for a given rotation 

 and mean density the homogeneous body is more oblate than the hetero- 

 geneous body, we are justified in concluding that probably instability first 

 appears in the figures of equilibrium for e = 0. 



Now consider the case of a heterogeneous compressible rotating fluid. 

 The preceding remarks pertaining to the existence of the figures of equi- 

 Hbrium still hold, though all the explicit defining equations are immensely 

 more complicated. But the question of stabihty has a new element in it. 

 As Jeans has shown in an important memoir,* gravitation itself becomes a 

 source of instability. It is easy to see that if for any reason there is a local 

 condensation in a compressible fluid, gravitation will tend to augment this 

 condensation. On the other hand, the elastic forces called into play tend to 

 destroy the condensation. The two kinds of forces are in conflict, and the 

 state of stability depends upon which will predominate when the equilibrium 

 is disturbed. Jeans showed that an infinitely extended nebula, of density 

 such that it possesses the properties of gases, is in unstable equiUbrium 

 independently of its mean temperature and density. That is, in this case it 

 is possible to introduce such local variations in density that the decrease in 

 gravitational potential energy shall more than balance the increase in the 

 potential energy of the elastic forces. 



There are as yet no quantitative results by means of which we may 

 measure the importance of this factor of instability. It is true that Jeans ^ 

 and Love * have proved the stabihty of the earth under certain assumptions, 

 but in view of its present existence, notwithstanding the many vicissitudes 

 which it has survived, this result may be taken as reflecting favorably upon 

 the method employed in obtaining it, rather than assuring us of the perma- 

 nency of our planet. In the absence of positive quantitative results we are 

 able to make only more or less probable hypotheses as to the importance 

 of this factor. 



There are seen to be two opposing factors entering into the question of 

 stability of heterogeneous compressible masses. As compared with homo- 

 geneous incompressible masses, the central condensation tends strongly 

 toward sphericity, as is shown both by theory and by the observed shape of 

 the sun and planets, and therefore presumably toward stability. But the 

 compressibility tends toward instability for local deformations and altera- 

 tions in density. We shall assume as appearing reasonable that in bodies 

 having strong central condensations and continuous changes in density these 

 two opposing factors approximately balance. To the extent to which this 

 assumption is justified, we may draw conclusions in regard to the actual 

 celestial bodies from what is known regarding the forms and stabiHties of 

 homogeneous fluids. For the purposes of additional safety in this procedure 

 we shall keep far from the conditions of possible disruption in the applica- 

 tions which follow. 



* The stability of a spherical nebula. <Phil. Trans., A, 199 (1902), pp. 1-53. 

 ' On the vibrations and stabihty of a gravitating planet. < Phil. Trans., A, 201 (1903), 

 pp. 157-184. 



» The gravitational stability of the earth. <Phil. Trans., A, 207 (1908), pp. 171-241. 



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