ME, j. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
31 
If Ave write y — 0 Ave pass to the case of a non-gravitating nebula, and Ave see 
that u x — 0 provided the ratio of pr 3 to XT remains finite at infinity. Noav Ave can 
keep the value of p and XT the same at every point by subjecting the nebula to 
an appropriate external field of force, and this field of force will be exactly the same 
as the gravitational field which Avas annihilated upon putting y — 0. It is spheri¬ 
cally symmetrical, and its potential vanishes at infinity to the order of 1/r, so that 
it comes within the scope of our previous analysis. For values of y intermediate 
between the natural value (y = l) and the value y = 0 we can obtain the same 
result by taking a field of force equal to 1 — y times the foregoing. As Ave increase 
y from 0 to 1 we obtain a linear series, in which the configuration of the nebula 
is unaltered, the nebula being gradually endoAved with the power of gravitation. 
For the general configuration of this series, consider the work done in a specified 
displacement, Avhich is proportional to S„ at every point. The potential (gravitational 
+ that of external field) after displacement Avill be of the form 
a + l>y$>,„ 
Avhere a and b are functions of r and independent of y. The total Avork done against 
this field during the displacement is therefore of the form 
By, 
Avhere B is independent of y and depends solely upon the particular displacement 
selected. The work done against the elastic forces is of course independent of y, 
and depends solely upon the displacement selected. This work is essentially positive. 
The total work is therefore of the form 
A + By, 
Avhere A is positive and B may (§ 2) be negative. Since y is proportional to u M this 
may be written 
A -j- B u x .(105). 
Suppose this function calculated for all possible displacements. Then we shall find 
that for values greater than some definite value of u x it is possible for the Avork 
done to become negative. For values of u x less than this critical value, the work 
will be positive for all displacements. Hence from the form of expression (105) 
it IoIIoavs that the passage of u x through a critical value denotes a real change from 
stability to instability, and that the stable configurations are given by the smaller 
values of u x . 
Recapitulation and Discussion of Results. 
§ 29. We haA-e seen that the vibrations of any spherical nebula may be classified 
into vibrations of orders n = 0, 1, 2, &c., a vibration of any order n being such that 
the displacement and change in temperature at any point are each proportional to 
