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Proceedings of the Royal Society of Edinburgh. [Sess. 
where M denotes the whole mass of the fluid. Thus we see that while the 
temperature and density both diminish to zero at infinite distance from the 
centre, the whole mass of the fluid is finite. 
§ 30. It is both mathematically and physically very interesting to pursue 
our solutions beyond ic — 5, to larger and larger values of k up to k = oc : 
though we shall see in § 43 below, that, for all values of k greater than 
3 (or &<1£), insufficiency of gravitational energy causes us to lose the 
practical possibility of a natural realisation of the convective equilibrium 
on which we have been founding. But notwithstanding this large failure 
of the convective approximate equilibrium, we have a dynamical problem 
of true fluid equilibrium, continuous through the whole range of k from 
— 1 to — oo, and from -f oc to 0 ; that is to say, for all values of k from 
0 to cc. In fact, looking back to the hydrostatic equation (10), and 
the physical equations (1), or (7), and (16), we have the whole 
foundation of equations (17) ... . (26), in which we may regard t 
merely as a convenient mathematical symbol defined by (6') in § 4. 
Any positive value of k is clearly admissible in (1), if we concern 
ourselves merely with a conceivable fluid having any law of relation 
between pressure and density which we please to give it, subject only 
to the condition that pressure is increased by increase of density. 
It is interesting to us now to remark, what is mathematically proved 
in § 44 below, that, unless k> 1J, 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 k has finite values greater than 
5, we leave it for the present and pass on to k = oo, or k = 1. In this case 
equation (1) becomes 
P_JP_ 
P V 
(31); 
which is simply Boyles law of the “ Spring of air,” as he called it. It was 
on this law that Newton founded his calculation 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 agreement 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 k = oc, our differential equation (24) fails; 
