STABILITY OF A C4RAVITATING PLANET. 
171 
We have now found a closer approximation to the value of (j) than that which was 
given in § 3, and have obtained the additional information that instability first enters 
through a vibration of order n = 1, It must, however, be borne in mind that these 
results are only true of the special and somewhat artificial case specified in § 8. 
Comparison with the Case of a Spherical Nehula. 
§ 21. It will Ije seen that the general ai'gument of § 3 will apply to the case of a 
gaseous planet or nehula if X l)e taken to mean the pressure in the gas. In this case, 
however, the laws of distribution of density and pressure are not independent. If 
the gas is in conductive equilibrium throughout, tire planet or nebida must be 
supposed to extend to Infinity, and for these conditions the criterion of stability 
was worked out in the former paper already referred to. Calling the elasticity 
of the gas k, the first point of bifurcation was found to l)e reached wlien the function 
Lv attains a certain finite value. Now Lf- p vanislies in comparison with p^, 
the mean density, so tliat writing a for the radius of the nebula, and X,, for the mean 
pressure (X,j = xpf, we have, at this first point of bifurcation 
27TpQklf\Q = CO . 
Comparing this with the general result obtained in § 3, we see that in this extreme 
case the value of (f) becomes infinite. This result is only of Importance to the present 
investigation as showing the tendency of a concentration of density about the centre. 
It seems to show that as the density becomes more concentrated about the centre, 
the value of cf) may be expected to Increase. XYe are therefore led to expect that in 
general <f) will have a value rather greater than that found for it upon the assumption 
of homogeneity of density. 
liECAPITULATION AND DISCUSSION OF RESULTS. 
§ 22. It will he well to recapitulate our results before attempting to draw any 
deductions from them. 
We consider a spherically .symmetrical mass of solid, liquid, or ga.seous matter. 
We denote the radius of this by a, the mean density by Pq, and the mean value of X 
by Xq, where X denotes an elastic constant or the pressure of the fluid, according as 
the matter is solid or fluid. We have seen that the stability of this dynamical .system 
depends upon the value of the function ypQa^l\, a pure number. When y = 0 (nc., 
when we deal with artificial matter which is totally devoid of gravitation) there can 
he no doubt that the system is stable. We have seen that a point of liifurcation 
occurs when the number ypu'«V^o ^ certain value f It has not been proved in 
the present paper that an exchange of stabilities accompanies this point of liifurcation, 
7 , 2 
