62 Trans. Acad. Sci. of St. Louis. 



^^ = -^^T^' (7) 



These equations are all consistent with each other. For 

 example, as will be at once seen, the value of P^ in (2) is 



CO 



P - IgddR, (9) 



r 



where d and g are given in (5) and (8). Also the value of 

 M^'m (3) is 



M^ = 4:7r iR-^ddR. (10) 



It will be assumed for the purposes of discussion that 

 R<B,. 



Let the entire mass contract, so that any mass originally 

 within a sphere of any radius r^, shall be within a radius r, 

 -iind assume also that 



^=§=,. (11) 



This means that the same law of density distribution shall 

 prevail in the second state as in the first. Assume also that 

 the temperature shall remain T^ throughout the mass. 



It is evident that the value of d in (5) is the reciprocal of 

 V, where v is the volume of unit mass and that from (5) and 

 (6) the product 



Pv= PX\=GT,. (12) 



o 



The same equation may also be applied to the entire sphere 

 in its initial condition. Let V^ be its volume. Its average 

 density is three times the density (5) at the surface. Then 



