454 



Mr. J. H. Jeans. 



'"The Stability of a Spherical Nebula," By J. H. Jeans, B.A., 

 Scholar of Trinity College, and Isaac Newton Student in the 

 University of Cambridge. Communicated by Professor G. 

 H. Darwin, F.E.S. Received June 15,— Eead June 20, 1901. 



(Abstract.) 



It is usual to take as the theoretical basis of the nebular hypothesis 

 the established fact that the equilibrium of a rotating mass of liquid 

 becomes unstable as soon as the rotation exceeds a certain critical 

 value. The present paper attempts to examine whether it is justi- 

 fiable to argue by analogy from the case of a liquid to that of a 

 gaseous nebula, and it is found that, on the whole, this question must be 

 answered in the negative. The paper is written with especial reference 

 to a paper by Professor G. H. Darwin,* in which it is shown that a 

 swarm of meteorites may, with certain limitations, be treated as a mass 

 of gas. The result obtained for a gaseous nebula can accordingly be 

 .at once transferred to the case of a meteoric swarm. 



It appears that the main difference between the stability of a liquid 

 .and that of a gas, lies in the difference of the parts played by gravita- 

 tion in the two cases. In the case of a liquid, gravitation is the factor 

 which supplies the forces of restitution ■ in the case of a gas these 

 forces are provided by the elasticity of the gas, while the influence of 

 gravitation, for some vibrations at least, tends towards instability. 



It is shown, in the first place, that the principal vibrations of any 

 spherically symmetrical nebula can be classified into vibrations of 



orders 0, 1, 2, oo , where a vibration of order n is such that the 



radial displacement and the cubical dilatation at any point are each pro- 

 portional to the same surface-harmonic S w of order n. 



The case of a nebula which extends to infinity is then examined, 

 . and it is shown that the stability depends solely upon the value of a 

 function as defined by 



t t 2irpr 2 



U "> = L K ' 



r — co K 



where p, k are the density and elasticity of the gas at a distance r from 

 the centre. Vibrations of zero order are of zero frequency ; vibrations 

 of order n (other than zero) become unstable as soon as u m exceeds the 

 value 



u x = J»(»+l). 



Hence instability enters first through a vibration of order n = 1, 

 and the nebula becomes unstable as soon as the value of u m exceeds 

 -unity. 



It is found that for a non-rotating nebula in which the gas equations 

 * ' Phil. Trans.,' A, vol. 180, p. 1. 



