210 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS 



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where 



p,=a,Hl+c) 2 (98) 



c = 3-aiyn2 



whence 



^ = l+c (99) 



Pi 



Then since the surface density is not to change, the total radius r/ and the 

 parameter a at any epoch are related by 



which, combined with (98), gives 



RcA^- (I +c) A -{-1=0 (101) 



a quadratic equation for A, in which are put 



The condition for reality of the roots shows that the maximum possible 

 radius is 



^/=| • ^ (102) 



as may be deduced also from the transformed equation 



From which it is seen that 72 as a function of A has a maximum value 



R = ^ ' at A = ,. y As A increases indefinitely from its smallest 



admissible value, which is r-— -, the mass also increases from zero without 



i -tC 



limit, but the total radius increases to the maximum value indicated and 

 then decreases toward zero. For any assigned radius less than the maximum 

 there would thus be two possible distributions of density, giving masses 

 less than and greater than the critical mass, and with mean densities less 

 and greater than double the surface density respectively. 



If, however, the constants a^, c be determined by (97), (98), from the 

 values of p^, p^ heretofore assumed for the earth, the computation ac- 

 cording to (94) shows that for some distance downward from the surface 

 the density would increase more slowly than with either of the formulae 

 used before, but then rise more rapidly and at the center reach a value of 

 15 to 16. In view of the probable corrections mentioned above as needed 

 to change the Laplacian formula into one agreeing better with the data 

 available, it appears that in comparison with (25) the formula (94) is much 

 less satisfactory, as an approximation to the general distribution of density 



