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



By means of this equation ( 39 ) takes the form 



da 2« ?>a 



— -f — —— =0 (43) 



dr r Ha 



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: — - p dT ' — ivdr, 

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

 stant pressure and ^Tis 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 



— Z v d T=zw?-dr=. 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 



>T rr = R 



vR, (45) 



J*i /»»• = it 



C p dT= iv | adr = 



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 = fig we should have 



: n dT=—w;3adr, (46) 



f 7 



: p l = -Wj3 a 



R 

 dr-w^vR (47) 







