32 Trans. Acad. Sci. of St. Louis. 



Thus, so long as the mass is a perfect gas the forms of the 

 density and temperature curves are rigorously the same after 

 contraction as before. If, however, the radiation were sud- 

 denly checked, say by surrounding the gaseous globe with a 

 solid shell impermeable to heat, the internal temperature 

 would soon become more equably diffused; the convective 

 currents would be interrupted or replaced by conduction of 

 some kind, and the nature of the temperature and density 

 curves would be rapidly altered. 



When the shell was removed, however, the original condi- 

 tions would return, and the curves of density and temperature 

 take their usual forms which satisfy the foregoing differ- 

 ential equations. From these considerations we conclude 

 that the distribution of temperature and density remain the 

 same, are represented by functions of the same form, which 

 satisfy the same differential equations so long as the radiating 

 mass is gaseous and condensing under conditions of convective 

 equilibrium. 



5. Elementary Derivation of the Law of Temperature. 



Suppose a gaseous globe of radius R and surface temperature 



T to be held in equilibrium by the pressure and attraction of 



its particles. Let P be the gravitational attraction exerted 



upon a thin layer of matter covering a unit surface of the 



globe, which may be regarded as the base of an elemental cone 



extending to the center. Then suppose the globe to shrink 



by loss of heat to a radius R. If the original element of mass 



now covered a unit surface the pressure exerted upon it would 



/R \ 2 

 thereby become P — P ( -5 I . But since the area of the ini- 



(R \2 



tial sphere surface has shrunk to £ = S I n J > the area of 



the elemental conical base into which the matter is compressed 

 has diminished in the same ratio. As the force of gravity is 

 increased while the area upon which it acts is correspondingly 

 decreased, it follows that in the condensed condition of the 

 globe the gravitational pressure exerted upon a unit area is 



-*.(§)■ 



The forces counterbalancing the increased 



