MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
39 
order of aA For small values of or, the value of or 1 will be proportional as we have 
seen (§ 31) to u x — 1 . so that this correction may be supposed to be proportional to 
iu w — l) 2 . The first points of bifurcation of orders 1 , 2 occur (in the spherical 
configuration) at — 1 = 9, 2^ respectively, where 6 is known to be less than 
Now it would seem to be fairly safe to neglect 0 2 , but even if we waive this point, it 
will be admitted that the correction of the order of (w« — I ) s cannot be so great as to 
change the order in which these two points of bifurcation will occur. 
We therefore see that a rotating nebula will become unstable for a comparatively 
small value of ad, the critical vibration being of order n = 1 . The new linear series 
is one in which (except for the spheroidal deformation caused by the rotation) the 
surfaces of equal density remain spheres, which are no longer concentric. The linear 
series of order n — 2 will accordingly be unstable : this is the analogue to the series 
of Jacobian ellipsoids in the incompressible fluid. 
§ 36. The case of a nebula which actually extends to infinity is much simpler. 
Here the value of u x is again unity, and this value is increased, as before, either by 
temperature-lag or rotation. Every point at which u x is greater than unity is in one 
sense a point of bifurcation, since starting from this point there is a series of 
unsymmetrical equilibrium configurations. Strictly speaking, these points do not 
indicate an exchange of stabilities, for the critical vibrations remain of frequency 
p = 0 even after passing the point. They possess, however, the property that a 
critical vibration, if once started, will continue increasing, since the forces of 
restitution (of whichever sign) vanish in comparison with the momentum of the 
vibration. 
§ 37. Let us now try and examine which of these two hypotheses is best capable of 
representing the “ primitive nebula ” of astronomy. Imagine a sphere S drawn in 
the nebula, the radius being a , and the pressure at this surface n. The matter 
inside S is to form a spherical nebula of finite extent, bounded by a sphere over 
which the pressure is tt, and this matter is to be of a density sufficient to warrant us 
in assuming the gas-equations at every point. The surface S will be continually 
traversed by matter, but this will be of no consequence if the losses and gains 
balance in every respect. The matter outside S must supply the pressure tt, and 
will also, as was explained in the introduction (§ 3), influence the matter inside S by 
its motion. 
Imagine the matter inside S to be executing a small vibration, and consider two 
extreme hypotheses as to the behaviour of the matter outside S. 
Suppose, in the first place, that the matter outside S is such that it and the 
matter inside S together form a perfect spherical nebula at rest. Then the motion 
of the matter outside S is given by the equations of vibration of such a nebula, and 
the influence of this matter upon that inside S is exactly that required in order to 
enable the matter inside S to execute the vibrations given by the equations of an 
infinite nebula. 
