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



M = ^£IS, (7) 



(8) 



These equations are all consistent with each other. For 

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



(9) 



where o and g arc given in (5) and (8). Also the value of 

 M^ in (3) is 



'P. 



M^ = 47r iB^ddB. (10) 



It will be assumed for the purposes of discussion that 



Let the entire mass contract, so that any mass originally 

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

 and assume also that 



!a=^ = P. (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 o 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^^=CT,. (12) 



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 



