704 Lord Kelvin : The Problem of 
to the present paper has calculated ¥(#) and ¥'(#)/¥ (a-), 
through the range from a?=oc to x='l. His results are 
shown in Table V. of the Appendix. Thus we niav consider 
¥(#) and its differential coefficient ^'(V) as known for all 
values of x through that range. 
§ 36. Using this solution, "^(x), instead of F in (39) above, 
we find that the solution of class A, which makes the central 
density C, is 
C^ 
(to) ( 4 °) ; 
and when we insert this expression for p in equation (38) we 
obtain 
. HS>) 
m Bcr 
E~^VC / -X • • • • ( 41 )- 
*(■*>) ' 
§ 37. From equations (40) and (41), with values of 
^("7c) and ^'("7c)/ XJr (~7c) obtained £rom the curves of 
"*P(x) and ^'(a^/M^a?) in the range from a? = co to a?=*l, and 
with the relation r— - where a is given 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 G 
increases by any small amount dC •; and to find also the 
changes in the radius of the spherical cell enclosing a given 
mass m, required to allow the mass to continue in equilibrium 
when the central density is increasing or diminishing con- 
tinuously. The following table shows the values of p or 
c *(7c> and An/EB ' or *t^>^*(-70)' fOT 
several of the larger values of r, corresponding to the central 
densities 1 and 1*21 respectively. 
