

484 MR. J. H. JEANS ON GRAVITATIONAL INSTABILITY 



sphere of particles both before and after displacement is exactly nil at a point inside 

 the sphere. Thus the gravitational field set up by a displacement of order 1 

 neutralises itself in a way not contemplated in the general argument outlined above. 

 Also the vibrations of order 1 and of greatest wave-length in the interior are not 

 available, for they represent solely a rigid body displacement. 



The question of vibrations of order 1 is treated in 3-7 ; it is shown that they 

 may be disregarded, and we pass to the consideration of vibrations of orders 2, 3, ..., 

 expecting (as, in fact, is found to be the case) that instability will first set in through 

 a vibration of order 2. 



It is only possible to discuss special cases, and the one which is most amenable to 

 analysis is that in which the pressure and density are connected by LAPLACE'S law, 

 p = c(p* <r 2 ). It is first proved ( 8-11) that, for a mass of such matter at rest, 

 the spherical form is stable for all displacements. Later ( 15-22) it is shown that 

 this is true when c varies inside the mass ; it is true even up to the case which is the 

 most likely to be unstable, in which the matter in the interior is of negligible density 

 and the main part of the mass is collected in a surface crust a sort of astronomical 

 soap-bubble. 



We proceed next to examine for what amount of rotation these figures will become 

 unstable, treating first the case in which c is the same throughout the mass. 

 Imagining c and a- to vary we can get a variety of types of mass. The surprising 

 result is obtained (by something short of strict mathematical proof) that the figure 

 which is the first to become unstable (as w 2 / 2?r P increases uniformly for them all) is 

 the perfectly incompressible one gravitational instability appears to act in the 

 unexpected direction, at any rate when the degree of rotation is measured by iv^feirp, 

 p being the mean density. As it was not possible to obtain strictly accurate figures 

 in this case, the result was checked by considering the artificial, but physically 

 analogous, problem of rotating cylinders of Laplacian matter, in which it was possible 

 to obtain perfectly exact results (14). The result was confirmed, and the additional 

 information was obtained that the value of w*/2-7rp remains surprisingly steady 

 through quite a wide range of compressibility (vide table on p. 471). 



The physical reason for this can, I think, be understood as follows. The more 

 compressible the matter is the more it tends to concentrate near the centre, i.e., in 

 just those regions where the " centrifugal force " obtains, so to speak, least grip on it. 

 Incompressibility neutralises the gravitational tendency to instability, but tends to 

 compel the matter to place itself so that the rotational tendency to instability can 

 act at the best advantage. 



The similar problem is next investigated ( 23, 24) when c varies inside the mass ; 

 in particular, the limiting case of a soap-bubble-like mass is considered. Again the 

 surprising result emerges that the value of nffe-wp needed to establish instability of 

 the symmetrical configuration is just about the same as before (vide table, p. 481). 

 The matter is now constrained to remain, so to speak, on the rim of a fly-wheel where 



