274 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which we may immediately verify by substitution in (32) (compare § 25 
above). The particular solution F must belong to one or other of the two 
classes, class A and class B, defined in § 25 above. 
§ 35. We shall denote by ^(x) what F(cc) of § 34 becomes, when the 
particular solution of (32), denoted by F, is of class A, with units so 
adjusted as to make x F(oc) = 1 ; that is to say, central density unity. Mr 
Green in his Appendix to the present paper has calculated ^(x) and 
s ^\x)/^f'(x), through the range from x = oc to x=l. His results are shown 
in Table V. of the Appendix. Thus we may consider ^(cc) and its differential 
coefficient ^\x) as known for all values of x through that range. 
§ 36. Using this solution, 'F(cc), instead of F in (39) above, we find that 
the solution of class A, which makes the central density C, is 
and when we insert this expression for p in equation (38) we obtain 
Bo* JL 
e 2 JC 
w 
* 
& 
( 40 ); 
( 41 ). 
§ 37. From equations (40) and (41), with values of ^ and 
-jq ^ -jq ^ obtained from the curves of dhjP) and ^\x)!^{x) in 
the range from x = oo to x = ‘l, and with the relation r = - where cr is given 
x 
by (37) above, we can tell exactly the density at any point of a spherical 
mass of an ideal Boylean gas, and the mass of gas within each spherical 
surface of radius r, when the gas is in equilibrium under its own gravitation 
only, and has a density at its centre of any stated amount C. It is 
interesting to examine by means of these solutions the changes in p and m 
at any given distance from the centre when the central density C increases 
by any small amount dC ; and to find also the changes in the radius of the 
spherical shell enclosing a given mass m, required to allow the mass to 
continue in equilibrium when the central density is increasing or diminishing 
continuously. The following table shows the values of p or J^r), an( l 
€ 2 m/EBo- or -y=-J / ^ or severa ^ the l ar g er values of r, 
corresponding to the central densities 1 and 121 respectively. 
