28 
ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
and this vanishes at all points, including r = R i; in the case in which Rj is put equal 
to infinity, if 
(97). 
u, -- = o 
•=» r 
The condition that cc 0 /r n+1 shall vanish at every point would lead to a similar 
equation to be satisfied at the origin, if there were no core. If, however, we retain 
the core, it leads to the same equation as was found in § 22 (equation (81), when the 
core is held at rest). Thus, our present method of finding a position of limiting 
equilibrium has led to a result different from that obtained by the search for a 
vibration of zero frequency, in that equation (97) replaces equation (79). 
The value of g at infinity is given by equation (82); hence we have 
r = qo ? 
[ E ,!*- 1 + Ej *' - 
(9S). 
As before, the equation to be satisfied at r = R 0 determines the ratio of E x to E. : : 
equation (97) is therefore satisfied if the real parts of p and p' are each less than 
unity. Now p, p are the roots of equation (91), hence this condition is satisfied 
provided 
n {n -f 1) < '2u x .(99). 
§ 2G. Let the kinetic and potential energies of a small displacement be given, in 
terms of the principal co-ordinates, by 
2T = apcp + apcp + . . . 
2V = bpc p -f- bc>Xcf + . . . 
so that the equations of motion are 
ape i — b x x — 0 
&c., and _p 2 is given by 
a Y p~ = b l .( 100 ). 
The method of §§ 20-24 amounted to finding vibrations such that p 2 = 0, and 
therefore, by equation (100), solutions of 
Zq = 0.(101). 
In § 25, on the other hand, we started with the supposition that the nebula extended 
to infinity, so that all the quantities a and b are liable to become infinite. The 
equation giving vibrations of frequency p — 0 is no longer equation (101), but is 
11 , =00 % 
and this is obviously more general than equation (101). 
(102), 
