212 PROFESSOR A. E. H. LOVE ON THE 



value of X+2/A for such a body would not be very different from the critical value 

 obtained for X+2/i by treating the body as homogeneous, and paying attention to 

 those types of displacement only which are specified by spherical harmonics of the 

 first degree. 



37. If this conclusion is admitted, as I think it must be, it would follow that a 

 spherical planet with a spherically symmeti-ical distribution of density, and stable as 

 regards radial displacements, might be unstable as regards displacements of the type 

 in question ; and then it would tend to be displaced in such a way that the boundary, 

 or any concentric sphere, moves to a position in which its centre no longer coincides 

 with the centre of gravity, while the matter in a thin spherical layer becomes 

 condensed in one hemisphere and rarefied in the other. The density being in excess 

 in one hemisphere and in defect in the other, and the excess or defect at any point, 

 at a stated distance from the centre, being proportional to the distance of the point 

 from the bounding plane of the two hemispheres, the distribution of density may be 

 aptly described as " hemispherical," and the state of the body may be described as one 

 of " lateral disturbance." The concentration of density towards one radius, on which 

 the centre of gravity lies, has the effect of diminishing the potential energy of 

 gravitation, and this diminution may more than counterbalance the increment of 

 potential energy due to strain. The proved existence of a critical value for X + 2/A (in 

 the case of a homogeneous body) indicates that this state of things really can occur. 

 An illustration of the nature of a hemispherical distribution of density will be found 

 in 47, 48 below. 



38. The results found by JEANS (1903) in the solution of the problem of the 

 gravitating sphere subjected to an external field of force, which balances gravitation 

 throughout the sphere when it is at rest, may be compared with those obtained above 

 in the case where the gravitation is balanced by initial pressure. In JEANS' solution, 

 just as here, the modes of vibration are specified by the spherical harmonics which 

 enter into the expression for the dilatation ; and, in any normal mode, the formula 

 for the dilatation contains a single spherical harmonic, and the radial displacement 

 at any stated distance from the centre is proportional to the same harmonic. If the 

 degree of the harmonic exceeds zero, instability can occur for a sufficiently small 

 value of the resistance to compression, whatever the degree of the harmonic may be. 

 It is not restricted to the case where the degree is unity, as it is in our problem of 

 initial stress ; but the value of the resistance to compression required for instability 

 diminishes rapidly as the degree of the harmonic increases. Instability enters first 

 when the harmonic is of the first degree,* that is to say, for lateral disturbances. 

 The critical values of (Pa? are 6 '72 when v = and 5 '33 when v = \, the degree of 

 the harmonic being unity. Since these values are a little less than the critical values 

 found in the solution of the problem of initial stress, it may be concluded that the 

 effect of initial stress, as compared with that of an external field of force, is to 



* The question of radial instability was not considered by JEANS. 



