273 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
but we deal with the failure by expressing t in terms of p by (19), and then 
modifying the result by putting k = oc. We thus find 
d 2 lo gp _ . 
dx 2 x 4 ' 
where x = - 
r 
(32); 
<x denoting a linear constant given by (37) below. Equation (32) is the 
equation of equilibrium of any quantity of Boylean gas, when contained 
within a fixed spherical shell, under the influence of its own gravity, but 
uninfluenced by the gravitational attraction of any matter external to it. 
The value of a- might, but not without considerable difficulty, be found 
from (22) by putting k = oc. But it is easier and more clear to work out 
afresh, as in § 33 below, the equation of equilibrium of a Boylean gas, un- 
encumbered by the exuviae of the adiabatic principle from which our 
present problem emerges. 
§ 33. Let 
P = B p (33), 
where B denotes what we may call the Boylean constant for the particular 
gas considered ; being its pressure at unit density. According to our units, 
as explained in §§ 10, 11, 12, B is a linear quantity. The analytical 
expression of the hydrostatic equilibrium is 
dp = - gpdr . 
where [(16) repeated] 
m/? 
J dr 
3 j ^ fp 
0 0 
• (34), 
. (35). 
^ — E/e 2 — 5-6. e r 2 
Eliminating p> from (34) by (33), and multiplying both members by r 2 , we find 
- r 
yd lOg p 
o f •/ o e* m 
drr z p = — - 
dr ~ 5’6.e.B i H B E 
(36). 
Differentiating this with reference to r, and then transforming from r to x 
as in equations (21). . . (24) above, we find (32), with the following expres- 
sion for <x : — 
5-6 
= -s-eB . 
(37). 
The equation of the first and third members of (36) gives 
m Bo- d log p 
E ~ e 2 dx 
(38). 
§ 34. Let now p = F(a?) be any particular solution of (32); we find as a 
general solution with one disposable constant C, 
( 39 ), 
VOL. XXVIII. 
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