MR. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
26 
We conclude that, independently of the values of u at points inside the nebula, 
the smallest value of u x for which a vibration of zero frequency and of order n is 
possible is given by 
u x = l (n + Y) 3 .(92), 
or, for all orders, is given by 
= H .(93), 
the limiting vibration being of order n — 1. 
It ought to be noticed that for this limiting vibration equation (82) fails to 
represent the solution owing to /x and p! becoming identical. The true solutions for 
real, zero, and imaginary values of p. — /x' may be put respectively in the forms 
£ = Cj y/r sinh (T (p ~ /x') log eR}, 
^ — C x \/r log eR, 
£ = O v s/r sin j (/x — p) log eR j , 
in which C x and e are constants of integration. 
At infinity p vanishes to the order of 1/r 3 , so that dpjdr — — 2 pfr. The value of 
g for very great values of r is therefore (equation (50)) 
g = XT (— 2A + Ra). 
At the outer boundary a surface - equation (32) directs us to take B = 0. 
Following this out, we find that at infinity A is of the same order as g, and therefore 
becomes infinite to the order of y/r. Suppose, on the other hand, that we start by 
taking A = 0, so that B = g XT?’. The value of B now vanishes at infinity to the 
order of 1 /y/r, and the surface-equation (32) is satisfied by a motion which vanishes 
at infinity. It would therefore appear to be easier to satisfy the boundary conditions 
when ?■ is actually infinite than when r is merely very great. This result opens up a 
somewhat difficult question, which will be considered in the next section. 
Before passing on, we may consider in what way the results which have already 
been obtained will be modified, if we suppose the core of the nebula to be free to 
move in space, instead of being held fast. For the free nebula u x = 1, so that our 
results show that a free nebula will be stable if the core is supposed fixed. The 
same must therefore obviously be true when the core is free to move, since a motion 
in which nebula and core move as a single rigid body will not influence the potential 
energy. When the nebula is not free, fixing the core may be regarded as imposing 
