702 Lord Kelvin : The Problem of 
by increase of density. It is interesting to us now to remark, 
what is mathematically proved in § 44 below, that unless 
X">Lj, the repulsive quality in the fluid represented by k in 
equation (1) is not vigorous enough to give stable equilibrium 
to a very large globe of the fluid, in balancing the con- 
glomerating effect of gravity. 
§ 31. As to the range of cases in which te has finite values 
greater than 5, we leave it for the present and pass on to 
./c = co, or k = l. In this case equation (1) becomes 
H (31)! 
which is simply Boyle's law of the " Spring of air/' as he 
called it. It was on this law that Newton founded his calcul- 
ation of the velocity of sound, and got a result that surprised 
him by being much too small. It was not till more than a 
hundred years later that the now well-known cause of the 
discrepance was discovered by Laplace, and a perfect agree- 
ment obtained between observation and dynamical theory. 
But at present we are only concerned with an ideal fluid 
which, irrespectively of temperature, exerts pressure in 
simple proportion to its density. This ideal fluid we shall 
call for brevity a Boylean gas. 
§ 32. For this extreme case of tc — co , our differential 
equation (24) fails ; but we deal with the failure by express- 
ing t in terms of p by (19), and then modifying the result 
by putting k — co . We thus find 
d 2 logp p , (T /Q0 . 
— t-^ j - = — [ -t' where .t'=- . . . (32); 
ax* x r 
a 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 anv matter external to it. The 
value of a might, but not without considerable difficulty, be 
found from (22) by putting k — co. But it is easier and more 
clear to work out afresh, as in § 33 below, the equation of 
equilibrium of a Boylean gas, unencumbered by the exuvise 
of the adiabatic principle from which our present problem 
emerges. 
§ 33. Let -o /OQ v 
7> = Bp (o3), 
where B denotes what we may call the Boylean constant for 
