AND THE NEBULAK HYPOTHESIS. 483 



of vibrations than an incompressible one, and as the mass is only stable when every 

 vibration individually is stable, it might be thought that a compressible mass had more 

 chance of being unstable or would become unstable sooner than the corresponding 

 compressible mass. This has on the whole been found not to be the case, and ->n 

 looking through the analysis the reason can be seen. 



A vibration in a compressible mass may be regarded loosely as a system of waves ; 

 the distance from one point of zero displacement to the next may be regarded as a sort 

 of wave-length of the vibration. The stability or instability of a vibration depends on 

 which is the greater the gain in elastic energy or the loss in gravitational energy 

 when the vibration takes place. But as between a vibration of great wave-length and 

 one of short wave-length there is this important distinction : for equal maximum 

 amplitudes the gravitational disturbance caused by the disturbance of great wave- 

 length is much greater than that caused by the disturbance of short wave-length, since 

 the elements of the latter very largely neutralise one another. Thus the change in 

 gravitational energy is enormously the greatest for disturbances of great wave-length, 

 while it is easily seen that the changes in elastic energy are approximately the same. 

 It follows that if the mass becomes unstable it will be through a vibration of tin- 

 greatest possible wave-length, i.e., a wave-length about equal to the diameter of the 

 mass. This general prediction is amply verified in the detailed problems that have 

 been discussed. When we reflect that the vibrations of greatest wave-length are 

 exactly those which are common both to compressible and incompressible masses, we 

 see readily that, in this respect at least, compressibility is likely to make but little 

 difference. 



The vibrations of greatest wave-length are put in evidence, both in the compressible 

 and incompressible mass, by the displacement of the surface. A vibration in which 

 the displacement is proportional to a zonal harmonic P, may be thought of as having 

 a wave-length approximately equal to -a-a/ii. In accordance with the principle that 

 vibrations of great wave-lengths are most effective towards instability, we should 

 expect the lowest values of n to give the vibrations which first bacome unstable, and 

 this is, in fact, found to be the case. But here a very real distinction enters between 

 the compressible and the incompressible mass. In the incompressible mass vibrations 

 of order n = 1 are non-existent, the displacement being purely a rigid txdy displace- 

 ment ; in the compressible mass vibrations of order = 1 can certainly occur, and so 

 might reasonably be expected to be the first to become unstable. 



It is in point of fact known that the incompressible mass becomes unstable through 

 vibrations of orders 2, 3, ... in turn; it is found that the compressible mass also 

 becomes unstable through vibrations of orders 2, 3, ... in turn, the vibrations of 

 order 1 failing completely to produce instability. The reason for this apparent 

 anomaly can, I think, be traced in the following way. In a displacement of order 1 

 any spherical layer of particles will after displacement be spread uniformly . 

 another sphere excentric to the first. The gravitational force produced 



3 Q 2 



