13G 



Mr. J. H. Jeans. On the Vibrations and [Nov. 8, 



Ci On the Vibrations and Stability of a Gravitating Planet." By 

 J. H. Jeans, B.A., Isaac Newton Student and Fellow of 

 Trinity College, Cambridge. Communicated by Professor 

 G. H. Darwin, F.K.S. Received November 8, — Read Decem- 

 ber 4, 1902. 



(Abstract.) 



The first part of the paper deals with the vibrations and stability of 

 a gravitating elastic sphere. The matter is not necessarily homo- 

 geneous, but is arranged in spherical layers. It is pointed out that,, 

 in the classical investigation of the displacements produced in a gravi- 

 tating sphere by given surface-forces, the most important of the gravi- 

 tational terms is omitted. The effect of this omission is to necessitate 

 a correction, and this may entirely invalidate the solution when we are 

 dealing with spheres of the size of the earth or other planets. In fact, 

 it appears that for a gravitating solid of the kind we are discussing the 

 spherical configuration may be one of unstable equilibrium, the instability 

 being brought about by the gravitational terms in the manner already 

 explained in a former paper.* 



Let a be the radius of the sphere, and y be the gravitation constant ; 

 let p be the mean density, and A the mean value of A, one of the 

 elastic constants. A general argument shows that the spherical com 

 figuration will be stable or unstable according as 



yp -a- 



< or > (i), 



where is a pure number which must be comparable with unity. 



If we put in an artificial field of force we can imagine a spherical 

 configuration of equilibrium in which the density and elastic constants 

 have uniform values throughout. The artificial field of force is, of 

 course, equal and opposite to the gravitational field produced by the 

 matter of the solid. The stability or instability is determined by a 

 criterion of the form of (i), and (f> can now be calculated exactly. If 

 we suppose A to represent A + 2/x in Love's notation (m + n in that 

 of Thomson and Tait), we find the values 



<f> = 1*6, when /x = 0, 

 <f> = 1*3, when \x — A. 



The vibration through which instability first enters is one in which 

 the displacement at every point is proportional to a harmonic of the 

 first order. It appears probable that for spheres which are not homo- 



* "The Stability of a Spherical Nebula," ' Phil. Trans.,' A, vol. 199, p. 1. 



