MR. J. H. JEANS ON THE STABILITY" OF A SPHERICAL NEBULA. 
27 
a constraint which does no work; freedom of the core therefore tends towards 
instability. It will be proved in § 28, that a nebula is stable for values of u x which 
are less than the critical value, and unstable for values greater than this value. 
Assuming this for the moment, we see that a nebula in which the core is free to move 
must necessarily be unstable if has a value greater than ljr. 
If then, we start with a free nebula and imagine to gradually increase from 
u x = 1 upwards, the core being free, it follows that the nebula will first become 
unstable when u x reaches some value such that 
n > > i 
(94). 
§ 25. The nebula extending to infinity, let us attempt to find the displacement 
which will be caused by a small disturbing potential v n given by 
47T 
= 9; 
in + 
j + a r > 
1 ] r »+i 1 i 
s, 
(95). 
It is clear that the displacement required will be given by our equations if we 
include in V (equation (27)) the terms 
47r 
2 n + 1 
The equation replacing (42) may be transformed in the manner of § 15, and the 
resulting equations will be those of § 16, except that we must replace (52) by 
1 cl ,. 
yh dr ^ 
= a, — 
r=R, 
A (p — cr ,) 
— 1 
,-=R, 
. . . (96), 
and (53) by a similar equation. 
If a displacement can be found to satisfy these modified equations, the external 
disturbing potential which will be required to hold the system in this displaced 
position will be given by equation (94). Now the condition that this displaced 
position shall be one of limiting equilibrium is that this disturbing potential must 
vanish. To be more precise, v n must be such that the force derived from it vanishes 
at every point of the nebula. We must therefore have 
a 1 r n = 0 
at all points of the nebula, including r = E, x . Now (95) may be regarded as an 
equation giving cq in terms of Tt 1 . Taking p =cr L , as before, we find from (95) 
