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XI. Gravitational Instability and the Nebulir / / i// t othesi*. 



x 



By J. H. JEANS, M.A., F.R.S. 



Received October 22, Read Novemlwr 27, 1913. 



Introduction. 



1. A CONSIDERATION of the processes of cosmogony demands an extensive knowledge 

 of the behaviour of rotating astronomical matter. What knowledge we have is 

 based upon the researches of MACLAURIN, JACOBI, POINCARE, and DARWIX. These 

 researches refer solely to matter which is perfectly homogeneous and incompressible, 

 although it is, of course, known that the primordial astronomical matter must be far 

 from homogeneous and probably highly compressible as well. The question of how 

 far we are justified in attributing to real matter the behaviour which is found to be 

 true for incompressible and homogeneous matter is obviously one of great importance. 



2. There are d priori reasons for expecting that there will be wide differences 

 between the two cases. Consider first a sphere of homogeneous incompressible 

 matter devoid of rotation. This will be stable if every small displacement increases 

 (or, at least, does not decrease) its potential energy. The sphere has a number of 

 independent possible small displacements which can be measured by the number of 

 harmonics which can be represented on its surface. The spherical configuration is 

 known to be stable because it can be shown that every one of these displacements 

 increases the potential energy. 



Contrast this case with the corresponding one in which the matter is compressible. 

 The number of possible small displacements in this latter case is measured by the 

 sum of the numbers of harmonics which can be represented on all the sphencal 

 s^lrfaces inside the sphere. Let R be the radius of the outer surface ; let r, r 1 , r", ... 

 be the radii of all the spheres which can be drawn inside this outer sphere, 

 and let r, r' n , /' ... R, be the number of independent harmonics which can be 

 represented on these spheres. To prove that the sphere is stable it is now necessary 

 to prove that every one of the r B +?', + /',,+ ... R. possible displacements increases 

 the potential energy. If we argue by analogy from the case of an incompressible 

 sphere we are, in effect, merely considering R. of these displacements and neglecting 

 the much greater number r n + 1\ +r" n + .... Furthermore, in these neglected dis- 

 placements, the nature of the displacement is essentially different from that in the 



VOL. CCXIII.-A 507. 3 N >ii.h.d p.t.i 7 . rebm.rr a, 11.14. 



