36 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
suppose that at infinity the linear velocity approximates to a finite limit, so that we 
may write 
o) = 91 jr 
for all values of r greater than a certain amount.* 
So long as we are only concerned with configurations of equilibrium and vibrations 
of frequency p — 0 , the rotation may be allowed for by the introduction of a force of 
amount orr sin 6 per unit mass, acting perpendicular to the axis of rotation ; or, what 
comes to the same thing, by the introduction of a potential 
| (1 — P 2 ) forWr, 
or 
( 1 -P 2 ) V' 
where, for all values of r greater than a certain value, 
V' = f H 2 log r .(114). 
Let us examine separately the two effects arising from the two terms of this 
potential, beginning with the term — P 2 V'. There will in this case be a correction 
to be applied to all equations, and this correction will consist of the addition of 
a small term containing orP 3 . Let us suppose that all symbols which have so far 
denoted functions of r, denote in future the mean value of the corresponding 
quantities averaged over a sphere of radius r. For instance, p is no longer the 
density at distance r from the centre, but is the mean density over the sphere of 
radius r. The density at any point will be of the form p + urP , 2 p. : , where p, is 
a function of r. We may in every case equate the coefficients of different harmonics, 
and by equating the coefficients of terms which do not contain the terms &rP 2 , we 
shall obtain the same equations as were obtained in the case of a> = 0 , except that 
the meaning of every term is altered. 
The equations derived from the parts which do not contain w will suffice, as before, 
to determine p, so that the values of p are of the same form as before, except that 
the quantities involved have a slightly different meaning. Hence the stability 
criterion is still given by the value of the stability function u x ; while equation (107) 
* This particular law is chosen for examination because it leads most quickly to the required result. 
The case in which w vanishes at infinity more rapidly than 1 /r is covered by taking 9. = 0. Here, 
however, the angular momentum vanishes in comparison with the mass, and it is not surprising to find 
that a rotation of this kind does not affect the question of stabitity. The case in which w vanishes less 
rapidly than 1 jr is physically impossible, since it gives an infinite linear velocity at infinity, but may be 
theoretically included in the case of 9 = oo . 
Any special assumption about the value of w at infinity must, however, disappear when we turn to the 
case of a finite nebula (§ 26), in which may be appropriately supposed to correspond to the surface 
velocity wR,. 
