ME, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
29 
It will be noticed that the method of §§ 20-24 is the method which is mathematically 
appropriate to the case of a nebula enclosed in a surface maintained at constant 
pressure, while the method of § 25 is that appropriate to an infinite nebula. In the 
former case, a vibration of frequency p = 0 may represent a real change from stability 
to instability; in the latter case such a vibration leads to an adjacent configuration of 
equilibrium, and is, in this sense, a point of bifurcation, but does not denote a change 
in the sign of jr. 
The General Case of a Nebula extending to Infinity. 
§ 27. The method to be followed has been explained in § 18. The general 
differential equation is of the sixth order. Four solutions have definite limiting forms 
when p — 0 ; the remaining two take singular forms. The former have been examined 
in § 16 ; the latter are represented mathematically (p. 18) by functions which do not 
approach a definite limit as p approaches a zero value, and physically (p. 11 ) by 
systems of steady currents. 
There are six constants of integration, E i5 E 2 , E 3 , E 4 , E 5 , E 3 , of which the two last 
belong to the singular solutions. Let us suppose (as is always possible (p. 19)) that 
the ratios of these six constants are determined from five of the boundary-equations, 
that which is not used being the equation satisfied by £ at the outer boundary. This 
remaining boundary-equation now takes the form (cf. equation (56)) 
Ep/q (Lj) + Eoifq (Itj) fi- E 5 \f/ 5 (Itj) -}- Ep f (E, 1 ) = 0 ... (103), 
in which the four E’s are definite quantities. The four \f/s must have definite 
limiting values (zero and infinity being included as possible values) when 14 = 00 . 
Thus in equation (103) some terms must preponderate over the others. When the 
nebula is isothermal, these terms are the first two. Hence, when the nebula is not 
isothermal, it follows from the principle of continuity, that the same two terms must 
still preponderate, at any rate for some finite domain including the isothermal 
nebula. Otherwise it would be j:>ossible to change the stability or instability of 
a nebula by an infinitesimal change in the physical constitution of the nebula. 
Hence throughout this domain, equation (103) must reduce to its first two terms, 
i.e., must become formally the same as in the case of the isothermal nebula. But 
the solution for £(and therefore the functions i/q, \jj. 2 ), remain formally the same in the 
general case as in this particular case, and therefore the stability-criterion derived 
from equation (103) remains formally the same. 
It follows that whether the nebula is isothermal or not (provided always that the 
configuration lies within a certain domain of equilibrium configurations) the critical 
configurations are given by the two equations (92) and (99). 
