1902.] 



Stability of a Gravitating Planet. 



137 



geneous, but in which the density is greatest near the centre, the 

 values of </> will be greater than those just stated, but the critical 

 vibration will still be such that the displacement is proportional to a 

 first harmonic term. 



In the former paper, already referred to, the suggestion was put 

 forward that the instability of a nebular sun or planet, which, upon 

 the nebular hypothesis, is supposed ultimately to result in the ejection 

 of a satellite, may be largely brought about by a gravitational 

 tendency to instability of the kind just described. We take, for the 

 moment, an extreme hypothesis, and imagine that this agency is the 

 preponderating agency, and that the rotational tendency to instability 

 may be disregarded in comparison. We then find that when the ejec- 

 tion of a satellite is taking place, or has just been completed, the value 

 of yp 2 oc 2 /X must be nearly equal to <f>. 



Except for the changes which have occurred since the consolidation 

 of the planets, the solar system supplies material for testing this con- 

 clusion. When a number of planets of varying masses have thrown 

 off satellites, we find (upon our present extreme hypothesis) that the 

 masses ought to be proportional to the squares of the radii. It is 

 found that this law is approximately obeyed in the solar system. It 

 is further found that the absolute values of the masses and radii are 

 approximately such as would be expected. 



It is interesting to compare two extreme hypotheses, the first refer- 

 ring the phenomena of planetary evolution solely to rotational, the 

 second solely to gravitational, instability. Given the approximate 

 values of the density and elasticity of a planet, and the fact that this 

 planet has thrown off" a satellite ; then the former hypothesis leads to a 

 certain inference as to the angular momentum of the system, the 

 latter to an inference as to the radius of the primary. The former 

 leads to no inference at all as to the size of planets which are to be 

 expected — they are as likely to be of the size of billiard-balls as of the 

 size of the planets of our system — while the latter leads to no inference 

 as to the angular momentum of the system, but presupposes it to be 

 small. The contention of the present paper is that the inferences which 

 are drawn from the former hypothesis are not borne out by observation 

 on the planets of our system, while those which are drawn from the 

 latter are borne out as closely as could be expected. The true hypo- 

 thesis must of necessity lie somewhere between the two extremes 

 which we are comparing, but the evidence seems to show that it is 

 much nearer to the latter (gravitational) than to the former (rota- 

 tional). 



We next consider a number of questions connected with the figure 

 of the earth. It seems to be almost certain that the present elastic 

 constants of the earth are such that a state of spherical symmetry 

 would be one of stable equilibrium. On the other hand, if we look 



