178 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



from distance 7\ to distance r', striking the surface, and then being covered 

 up more and more deeply, settled finally to distance r. The diagram gives 

 an idea of the distribution of density at the instant of deposition of that 

 particle, A being the center, the part CD along the radius referring to the 



An B C 



homogeneous spherical shell which is being drawn up to supply surface 

 deposits at B, BC the empty region through which the particles are falling, 

 and AB the radius of the partially formed planet, the variation of density 

 within which is determined by the condition of equiUbrium under a definite 

 law of compressibility. 



Since the mass already deposited is m, the energy of impact per unit- 

 mass is under these special conditions: 



-Hhr) (i^> 



where m, r', r,, are to be thought of as functions of r; then the total energy 

 of impact is 



pjnQ--l-\r'dr (18) 



The quantities E and E^ are portions of the potential energy <2> — (P,, ex- 

 hausted during condensation of the homogeneous sphere to a condition of 

 density matching that of the completed planet, 0,^ being the value of for 

 a sphere of assigned mass m^ and uniform density p^, and determined by 



The remainder of the energy 0—0^, if any, is to be treated as of the third 

 kind named above; but in case the equilibrium in the mass already depos- 

 ited at each moment of the process is adiabatic, so that the two thermo- 

 dynamic curves mentioned are identical, it seems probable that this third 

 kind does not exist, and the total amount - 0j^ transformed is completely 

 accounted for as E + E^. This is later proved to be the case for one partic- 

 ular pressure-density law. In any case it is of course not meant that the 

 primitive distribution need be at all like that in the homogeneous sphere 

 mentioned, but any other supposition would modify only the energy of im- 

 pact, leaving the above point of view still useful as a check on the compu- 

 tation of the compressional portion E; this is verified by the expression of 

 Ef as the difference 



4nk I ^r'dr-^Ttk I ^r'dr 





