and x / HT* 



290 Sir William Thomson on the Equilibrium of 



which, if we put p i_1 = u (H)> 



1/(A-1) = « (12), 



V J-i) =i (13) 



takes the remarkably simple form 



3?=-;? (14) - 



Let fix) be a particular solution of this equation ; so that 



/"(*)-■- L/W]"*-« 1 



and therefore L . . (15). 



f'(mx) = - [f(mx) ] K m-%" 4 J 



From this we derive a general solution with one disposable 

 constant, by assuming 



u=Of(mx) (16); 



which, substituted in (14), yields, in virtue of (15), 



m 2 = 0- K + 1 ...... (17); 



so that we have, as a general solution, 



u=Gf[xQ-^~ l) -\ (18). 



Now the class of solutions of (14) which will interest us 

 most is that for which the density and temperature are finite 

 and continuous from the centre outwards, to a certain 

 distance, finite as we shall see presently, at which both vanish. 

 In this class of cases u increases from to some finite 

 value, as x increases from some finite value to c© . Hence if 

 u=f(x) belongs to this class, u = Qf(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 ^(/e-— l)th powers 

 (or | 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 dif- 

 ferent masses). Thus, for example, one sixteenth central 

 temperature corresponds to eight-fold diameter : one eighty- 

 first central temperature corresponds to twenty-seven fold 

 diameter. 



* This adjective excludes stars or nebulas rotating 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. 



