20 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
Y a term a 0 r n . Equation (39) gives (except in the special case of c = constant), B = 0. Equations (50) 
and (51) remain unaltered, and give a solution of the form 
A = Ei/i (r) + Eo/ 2 (?-). 
Now we must have A = 0 when r = R 0 , and this determines the ratio Ei/E 2 . Also equation (49) must be 
satisfied, and this leads to a second and different value for Ei/E 2 . 
A second example, of less interest but greater simplicity, will perhaps help to elucidate the matter. 
Imagine a non-gravitating medium in equilibrium under no forces inside a rigid boundary. Let the law 
connecting pressure and density for any particle be w = up, where k varies from particle to particle. In 
equilibrium vr has a constant value ~ 0 . Suppose now that we attempt to find an adjacent configuration 
which is one of equilibrium under a small disturbing potential Y. The general equations of equilibrium 
are three of the form 
dV _ 1 dvr 
do: p dx 
If the position of equilibrium only varies slightly from the initial position, dwfdx will be a small quantity 
of the first order, so that (to the first order of small quantities) p may be replaced by its equilibrium value 
vt 0 /k. We now have 
dvr _ vtq dV 
dx k dx ’ 
and therefore, since w is a single-valued function of position, 
f 1 dV , A ,. x 
" . 
the integral being taken along any closed path. Since Y and k are absolutely at our disposal, this 
equation is, in general, self contradictory. What we have proved is that there will only be an “ adjacent ” 
configuration of equilibrium under a potential Y if V is a single valued function of k, a condition which 
will not in general be satisfied by arbitrary values of Y and k. 
It is not difficult to see the physical interpretation of this last result. There were initially an infinite 
number of equilibrium positions, and therefore an infinite number of vibrations of frequency p = 0. To 
arrive at the configuration of equilibrium under the disturbing force we must imagine vibrations of 
frequency p = 0 to take place until equation (i.) is satisfied; the disturbed configuration will then differ 
only slightly from the configuration of equilibrium. For instance, if the disturbing field of force consists 
of a small vertical force g, the fluid must be supposed to arrange itself in horizontal layers of equal density, 
before we attempt to find the disturbed configuration. 
The interpretation of the result obtained in the first instance is similar, but more difficult. Consider 
a linear series of equilibrium configurations, obtained by the variation of some parameter a, such that the 
spherical configuration of our example is given by a = 0. The other configurations are not symmetrical, 
the asymmetry being maintained, if necessary, by an external field of force. Every degree of freedom in 
the configuration a = 0 must have its counterpart in the configurations in which a is different from zero. 
In particular, the principal vibrations of § 12, in which (for the configuration a = 0) the dilatation, normal 
displacement, and temperature-increase all vanish, must have counterparts for all values of a. But when 
a is different from zero, the above three quantities cannot be supposed to all vanish. In general, therefore, 
these degrees of freedom provide solutions of the volume-equations, and these solutions contribute to the 
boundary-equations. In the special case of a = 0, these solutions do not affect the boundary-equations at 
all, so that to rectify the boundary-equations we must, so to speak, take an infinite amount of these 
solutions. In other words, the complete vibration of frequency p = 0 becomes identical with one of the 
vibrations of § 12, in which u, A, and T x all vanish. 
