186 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



Comparison with (34) and (35) gives the fundamental relation 



E + Ei = (44) 



showing that under the conditions here assumed the original store of poten- 

 tial energy is entirely accounted for as transformed by impact and static 

 compression. The proof refers to a primitive condition of infinite disper- 

 sion, but a similar conclusion would hold for any initial configuration and 

 distribution of velocities provided stands for the entire primitive store of 

 energy, potential and kinetic, a variation in which would make an equal 

 change in E^, but leave E unaltered. 



As represented by the preceding equations the character of the process 

 of accretion, upon a nucleus composed of material of definite compressibihty, 

 implies that the deposition of a new layer of given thickness brings about a 

 certain increase of compression of the nucleus and a corresponding sinking 

 of the former surface toward the center; only part of the thickness of the 

 new stratum is thus effective in producing actual increase of geometric 

 dimensions. In order to specify this differential depression numerically, let 

 a factor of depression D be defined as the ratio, to the total thickness of 

 stratum, of the part which sinks below the level of the former surface; 

 then 1 - D is the ratio of geometric increment of radius to total thickness of 

 stratum. 



For simplicity of notation let /? now stand for the angle equivalent of 

 the radius of the nucleus at a given epoch, the mass being by (25) and (26) : 



^=^{i9-/3^cot/3| 

 where /?, is the fixed density of surface rock, so that 



c;7n=^|(l-/3cot^)2 + i?4d/? 



dm being the increment of mass and d^ the increment of radius in the angu- 

 lar units defined. If, however, a stratum of the same mass dm were laid 

 down without producing compression of the nucleus beneath, the relation 

 to the total thickness d^ of the stratum would be given by 



From these follows for the depression factor 



D = l-t = 



1 



d^ /^sin /?y (45) 



1 + 



{^) 



where C is defined as in (36). The factor D is tabulated in column 6 of 

 table 3, in terms of /?/, which there represents the momentary radius of the 

 free surface. 



