MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 35 
d 2 
In the more general case in which — log (XT) cannot be supposed to vanish, it is 
clear that this term will vanish at infinity in comparison with the other terms 
in (ill), if r] has the limiting value given by (112), and therefore that (112) is the 
limit, at infinity, of the solution of ( 108 ). 
Of the two arbitrary constants, A and B, the former corresponds to the indeter¬ 
minateness of the linear scale upon which the nebula is measured, the second to the 
indeterminateness of the conditions at the inner surface of the nebula. If there is no 
core, there is only one value of A/B which will give a finite density of matter at the 
centre of the nebula. Further information as to equilibrium configurations can be 
found in Professor Darwin’s paper, or in a paper by A. BitterA 
For our purpose it is sufficient to know that the second term in y vanishes with x 
for all values of A and B. Hence at infinity 
V = 
V" , , A 4 TU 2 
- -I- loo’ — 
2\ 2 T 2 “ 2tt ’ 
\ 3 T 3 A 
2 ir 
^,V"/2A 2 T 2 
XT 
27rr 2 
gV"/'2A 2 T 2 
? 
and hence (equation ( 107 )) 
u 
00 
Npl~ -V ,, /2A 2 T 2 
AT 6 
1 + 
V" 
2\ 2 T 2 
(113). 
Putting V" = 0, we arrive at the anticipated result that the stability function has 
a unit value, for every nebula which extends to infinity in such a way that XT 
has a finite limit at infinity. 
A Slowly Rotating Nebula. 
§ 31 . The case which is of the greatest physical interest, is that in which the 
nebula is not at rest but is rotating in a position of relative equilibrium 
Here the arrangement is no longer in spherical shells, so that the foregoing analysis 
breaks down. If, however, we suppose the rotation w to be so small that cA may be 
neglected, it will be easy to modify the foregoing analysis, so as to take account of 
rotation. 
We shall still suppose the nebula to extend to infinity, so that we must not suppose 
the rotation to be the same at all distances, for in this case a finite value of o> would 
imply an infinite velocity of those parts of the nebula which are at infinity. Let us 
* ‘ Wied. Ann./vol. 16, p. 166. 
F 2 
