42 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
The functions p x , p. 3 . . . . are to be determined from the condition that V and p 
shall satisfy the three equations of equilibrium, which are of the form 
da _ d\ 
p dx dx 
An Isothermal Nebula. 
§ 40. Let us suppose, for the sake of simplicity, that the nebula is at uniform 
temperature, and extends from r = 0 to r = oo . We have already seen (equation (77)) 
that the critical vibration for a nebula initially isothermal, is one in which the nebula 
remains isothermal. Hence it follows that ifi a nebula changes its configuration 
through coming to a point of bifurcation, when moving on a series of isothermal 
and spherical configurations, then the new series will also be one in which the 
equilibrium is isothermal. 
We may now write ct = «p, where k is a constant, and the three equations of 
equilibrium become equivalent to the single equation, 
or 
k log p — N + c , 
V+c 
p = e K 
(118). 
Now the series in question is, as we have seen, approximately represented, near to 
the point of bifurcation, by taking only two terms of (115), and consequently only 
two terms of (116). In this case equation (118) becomes : 
Po + PiJ'i — e j 1 + jL + i („’ Pi) + 
. . (119). 
Equating coefficients, we find that p 0 is given by the equation 
0,1+r 
Po = e '« , 
the same equation as in the case of' perfect spherical symmetry. Also p x is given by 
the equation 
Pi - Ji¬ 
lt will be easily verified that this equation is exactly equivalent to our former 
equation (38). The equation contains an arbitrary multiplier in its solution. This 
may be taken to be a, the parameter of the series, so that we may write 
Pi = «oq, 
