See — Temperature of the Sun and Ages of Stars and Nebulae, 27 



By means of this equation ( 39 ) takes the form 



da 2a S(T ^ ^ 



7+ 7^_=0 (43 



f/r r Ma 



A well-known theorem in the Kinetic theory of gases states 

 that the internal heat of any element in convective equilibrium 

 is equivalent to the mechanical energy required to raise the ele- 

 ment to the limits of the atmosphere ; for the adiabatic com- 

 pression of the element from infinite expansion would develop 

 this amount of heat ; or an equivalent work would be done by 

 the particles if the mass were allowed to expand indefinitely, 

 as happens when the element circulates from a depth below 

 the surface to the limits of the atmosphere. 



Thus if w denote the caloric equivalent of a kilogram- 

 meter, and dr the height of the atmosphere, we shall have the 

 following differential relation between the internal heat and 

 gravitational work upon a kilogramme of air: — CpdT = wdr, 

 in which as before Cp is the specific heat of the gas under con- 

 stant pressure and dT is the change of absolute temperature. 

 When the kilogramme of air is elevated above the surface dr, 

 where the force of gravity is g', we shall have 



— ^^d T=w ^di' = wadr ( 44 ) 



The total amount of heat given up by the element in 

 ascending from the center of the sphere to the surface will be 

 given by 



J.T nr = R 



:jfiT=w Xadr^vR, (45) 



To *^r = 



where v is a small numerical coefficient, which must be 

 found by successive approximations. If the force of gravity 

 at the surface of the sphere were G = l^g we should have 



:^dT=—w^adr, (46) 



^pT^ = — w;^ I adr = lo^vR (47) 



•^0 



