.(,-£}--<*** w> 



a Gas under Us own Gravitation only. 289 



elementary hydrostatics, 



f r =-p(M+£dr^p)p^ . . (6), 



whence d 



dr [ 



where M denotes the whole quantity of matter within radius a 

 from the centre ; which may be a nucleus and gas, or may be 

 all gas. 



If the gas is enclosed in a rigid spherical shell, impermeable 

 to heat, and left to itself for a sufficiently long time, it settles 

 into the condition of gross-thermal equilibrium, by " conduc- 

 tion of heat," till the temperature becomes uniform through- 

 out. But if it were stirred artificially all through its volume, 

 currents not considerably disturbing the static distribution of 

 pressure and density will bring it approximately to what I 

 have called convective equilibrium* of temperature — that is to 

 say, the condition in which the temperature in any part P is 

 the same as that which any other part of the gas would acquire 

 if enclosed in an impermeable cylinder with piston, and dilated 

 or expanded to the same density as P. The natural stirring 

 produced in a great free fluid mass like the Sun's, by the 

 cooling at the surface, must, I believe, maintain a somewhat 

 close approximation to convective equilibrium throughout the 

 whole mass. The known relations between temperature, 

 pressure, and density for the ideal " perfect gas," when con- 

 densed or allowed to expand in a cylinder and piston of 

 material impermeable to heat, aref 



P = H.Tp* (8), 



t^Tp*- 1 (9); 



where k denotes the ratio of the thermal capacity of the gas, 

 pressure constant, to its thermal capacity, volume constant, 

 which is approximately equal to 1'41 or 140 (we shall take 

 it 1*4) for all gases, and all temperatures, densities, and pres- 

 sures ; and T denotes the temperature corresponding to unit 

 density in the particular gaseous mass under consideration. 



Using (8) to eliminate p from (7) we find 



r^cg^i-. ^ft-iu ao) . 



* See " On the Convective Equilibrium of Temperature in the Atmo- 

 sphere," Manchester Phil. Soc. vol. ii. 3rd series, 1862 ; and vol. iii. of 

 Collected Papers. 



t See my Collected Mathematical and Physical Papers, vol. i. 

 Art. xlviii. note 3. 



Phil. Mag. S. 5. Vol. 23. No. 142. March 1887. X 



