40 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
Suppose, next, that the matter outside S consists mainly of molecules or of masses 
of matter which are describing hyperbolic or parabolic orbits, or which come from 
infinity and after rebounding from the nebula return to infinity. Suppose, further, 
that the interval during which such a mass is appreciably under the influence of the 
matter inside S is so small that it is not appreciably affected by the motion of the 
latter. In this case the matter outside S may he regarded as arranged at random, 
independently of the vibrations of the matter inside S ; it will not, as under our first 
supposition, take up the motion of the matter inside S to any appreciable extent. 
Hence the matter outside S will exert no force upon that inside S except the 
constant pressure v, and the vibrations of the matter inside S wall be those of a 
spherical nebula of finite size, bounded by a surface at constant pressure n. 
These two extreme hypotheses lead, as w T e can now see, to the two conceptions of a 
nebula put forward in § 4. In nature the truth will lie somewhere between these 
two hypotheses, and it is by no means easy to decide which of the two gives the 
better representation of an actual nebula. We shall, however, be within the limits 
of safety if we assert of an actual nebula only those propositions which are true of 
both our ideal nebulae. 
§ 38. We may accordingly sum up as follows :— 
(i.) A nebula at rest and in absolute equilibrium in a spherical configuration will 
always be stable. 
(ii.) Such a nebula may become unstable as soon as the temperature-lag is taken 
into account. 
(iii.) There will be a linear series of configurations of relative equilibrium of a 
rotating nebula, starting from a non-rotating spherical nebula (supposed 
stable), and such that the configuration is symmetrical about the axis of 
rotation. This linear series corresponds to the series of Maclaurin 
spheroids. 
(iv.) The first point of bifurcation on this series occurs for a comparatively small 
value of the angular rotation. 
(v.) The second series through this point is one in which the configurations 
possess only two planes of symmetry. Initially the configuration is such 
that the equations to the surfaces of equal density contain only terms in 
the first harmonic in addition to those required by the angular rotation. 
(vi.) There is a linear series which corresponds to the series of Jacobian ellipsoids, 
each configuration possessing three planes of symmetry. The point of 
bifurcation at w^hich this series meets the series mentioned in (iii.) is a point 
at which the angular rotation is much larger than that at the point of 
bifurcation mentioned in (iv.). 
(vii.) This latter linear series appears to be always unstable. 
