1887.] Sir W. Thomson on the Equilibrium of Gas. 
115 
Let / (x) he a particular solution of this equation ; so that 
/"(*)= -[/wr® -4 ) 
and therefore V. . . . (15y. 
f'(inx)— - \_f(mx)\ K m~ A x~ ^ ) 
From this we derive a general solution with one disposable 
constant, by assuming 
u — Cf(mx ) (16); 
which, substituted in (14), yields in virtue of (15), 
m 2 = c -*+i (17) ; 
so that we have, as a general solution, 
^ = C/^C-^->] (18). 
Now the class of solutions of (14) which will interest us most is 
that for which the density and temperature are finite and con- 
tinuous from the centre outwards to a certain distance, finite as w 
shall see presently, at which both vanish. In this class of cases u 
increases from 0 to some infinite value, as x increases from some 
finite value to oo . Hence if u = f(x) belongs to this class, u = Cf(mx) 
also belongs to it; and (18) is the general solution for the class. 
We have therefore, immediately, the following conclusions : — 
(1) The diameters of different globular* gaseous stars of the 
same kind of gas are inversely as the J(»<-l)th powers (or f 
powers) of their central temperatures, at the times when, in the 
process of gradual cooling, their temperatures at places of the same 
densities are equal (or “ T ” the same for the different masses). 
Thus, for example, one sixteenth central temperature corresponds to 
eight-fold diameter; one eighty-first central temperature corresponds 
to twenty-seven fold diameter. 
(2) Under the same conditions as (1) (that is, H and T the same 
for the different masses), the central densities are as the Kth powers 
* This adjective excludes stars or nebul;e rotadmg steadily with so great 
angular velocities as to be much flattened, or to be annular ; also nebulae 
revolving circularly with different angular velocities at different distances 
from the centre, as may be approximately the case with spiral nebulae. 
It would approximately enough include the sun, with his small angular 
velocity of once round in 25 days, were the fluid not too dense through a large 
part of the interior to approximately obey gaseous law. It no doubt applies 
very accurately to earlier times of the sun’s history, when he was much less 
dense than he is now. 
