MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
21 
An Isothermal Nebula. 
§ 20. Let us now examine the form assumed by our equations in the simple case in 
which X and T are the same at all points of the nebula. We find that, considering 
only the equations for the case of p — 0, equation (39) reduces to 
C = 0.(77), 
and, in virtue of this simplification, the equation of conduction of heat (36), and the 
two thermal boundary conditions (33 and 34) are satisfied identically. We are left 
with equation (55) to be satisfied throughout the gas, and equations (32), (35), (52), 
and (53) to be satisfied at the boundaries. 
The solution of equation (55) is given in equation (56). Now we must satisfy 
equation (32) by taking B = 0 at r = R x , and this, by equation (50), gives the value 
of A at r = Rj in terms of E 2 and E 2 . Hence equation (52) reduces to a homo¬ 
geneous linear equation between E : and E 3 . 
When n is different from unity, we satisfy equation (35) by taking A = 0 at 
r = R 0 , and this reduces equation (53) to a homogeneous linear equation between 
E : and E 2 . 
When n — 1, equation (35) reduces to a linear equation between (A) r = Ro , E x and E 3 . 
Equation (53) is a second equation of the same form, and the elimination of (A),. = Eo 
from these two equations leads to a homogeneous linear equation between E : and E 2 . 
Thus, in either case, we see that the whole system of equations reduces to a pair 
of homogeneous linear equations between E : and E 2 . The elimination of these 
quantities leaves us with a single equation between n and the constants of the 
nebula. 
We can, therefore, satisfy all the equations for a vibration of frequency p — 0 by 
imposing a single relation upon the constants of the nebula. The unknown solutions 
which are multiplied by E 5 and E 6 have not been taken into account at all, but since 
the condition that there shall be a vibration of frequency p = 0 must of necessity 
reduce to a single equation, it will be clear that if these solutions had been taken 
into account, we should have found it necessary to take E 5 = E 6 = 0. 
Thus, in the case which we are now considering, a vibration of frequency p — 0 
is equivalent to a configuration of limiting equilibrium. It is not hard to see that 
this results from the fact that the particles of which the nebula is composed are 
physically indistinguishable. This very fact, however, introduces a further complica¬ 
tion into the question. It will be noticed that, although the value of £ has been 
found at every point of the nebula, it is impossible to determine the separate values 
of A and B. On the other hand, the physical vibration must have a definite limiting 
form when p — 0. Now it is easy to see that a motion of the gas in which 
£ vanishes at every point of the gas, and in which A and B vanish separately at the 
