12 
MR. J. H. JEANS ON THE STABILITY' OF A SPHERICAL NEBULA. 
constants of integration, and the values of these quantities will be wholly real if i r p is 
real. The boundary-equations enable us to determine the constants of integration 
and also provide an equation for ip. Every term in these equations will be real if ip 
is real. Hence the frequency equation can be written in the form 
f(ip) = 0, 
where f(x) is a function of x in which all the coefficients are real, these coefficients 
being functions of n and of the quantities which determine the equilibrium configura¬ 
tion of the nebula. 
It follows that the complex roots of ip will occur in pairs of the form 
ip = 7 ± i S, 
where y and 8 are both real. There may also be roots for which ip is purely real, so 
that 8 = 0, and y exists alone. 
The vibration corresponding to any root is stable or unstable according as y is 
negative or positive. 
If the equilibrium configuration of the nebula changes in any continuous manner, 
so as always to remain an equilibrium configuration, the values of ip will also change 
in a continuous manner, and for physical reasons these values can never become 
infinite. Hence, if the configuration of the nebula changes from one of stability to 
one of instability, it must do so by passing through a configuration in which there is 
a vibration for which y = 0. 
§ 14. For the present we shall not discuss the actual stability or instability of any 
configuration, but shall examine under what circumstances a transition from stability 
to instability can occur. 
We therefore proceed to search for configurations in which there are vibrations 
such that y — 0. Now for such a vibration we have either a root of the frequency 
equation p = 0, or else a pair of roots of the form ip — ffi i 8. 
In the latter case the corresponding vibration is one in which a dissipation of energy 
does not occur. A necessary condition for such a vibration is that no conduction of 
heat shall take place. Hence both sides of the equation of conduction of heat 
(equation 31) must vanish. Excluding adiabatic motion (represented by the 
vanishing of the factor MpB -f- C*.C), this condition compels us to take 
P 
= 0 
together witl 
chc dC k [ d / 0 dC\ . i \ ru i 
dr d i 
dr I 
a 
j 
d/C\ dT 
K 
dr \2T di 
— K 
dT f 
Id/ 
4 dA n (n + 1) — 2 1 
dr\ 
p dr \ 
. dr J 
i 
1 
VJ 
= 0 
(3G). 
