4 
MR. J. H. JEANS OX THE STABILITY OF A SPHERICAL NEBULA. 
consisting' either of molecules or meteorites, so that we are now called upon to take 
account of the gravitational forces exerted upon the nebula by this matter. This 
whole question is, however, deferred until a later stage; for the present we turn to 
the purely mathematical problem of finding the vibrations of a mass of gas which is in 
equilibrium in a spherical configuration. We shall consider two distinct cases. In the 
first, equilibrium is maintained by a constant pressure applied to the outer surface of 
the nebula, this surface being of radius Xi j. In the second, the nebula extends to infinity, 
and it is assumed that the ordinary gas equations are satisfied without limitation. We 
suppose for the present that the gas is in thermal equilibrium throughout. It is not, 
however, supposed that the gas is all at the same temperature ; to make the question 
more general, and to give a closer resemblance to the state of things which may be 
supposed to exist in nature, it will be supposed that the gas is collected round a solid 
spherical core of radius H 0 , and the temperature will be supposed to fall off as we 
recede from this core to the surface, the equation of conduction of heat being satisfied 
at every point. We shall also suppose that the gas is acted upon by an external 
system of forces, this system being, like the nebula, spherically symmetrical. The 
reason for these generalisations will be seen later; it will at any time be possible to 
pass to less general cases. 
The Criterion of Stability. 
The Principal Vibrations of a Spherical Nebula. 
§ 5. We shall take the point about which the nebula is symmetrical as origin. It 
will be convenient to use rectangular co-ordinates x, y, z, in conjunction with polar 
co-ordinates r , 0, <f>. We shall imagine the nebula to undergo a small continuous 
displacement ; let the components of this be y, £, when referred to rectangular 
co-ordinates, and u, rv, nv sin 0 when referred to polars. Thus the point initially at 
IP 2 or r, 0, $ 
is found after displacement at 
a; + £ y + y, z + £ or r + u , o + V, 4> + U\ 
The cubical dilatation of this displacement will be denoted by A. so that 
°ii f i 
6y 8: 
k-(hV) + 
cr x 
1_ 0_ 
sin 0 c6 
(r sin 0) + 
CIO 
dfi 
