184 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



densed by deposition of the remaining layers, so that in order to determine 

 the total amount and distribution of the energy of impact it is necessary to 

 know how the compression advances in the existing nucleus as material is 

 deposited at its surface. 



HISTORY OF THE COMPRESSION. 



It is possible, under the foregoing assumptions, to trace such a history 

 of the accretion, if it be supposed that the equilibrium of the mass is main- 

 tained at each stage of the process. For if the preceding equations can be 

 accepted as approximate representation of the action of the earth-substance 

 under compression, to a similar approximation equation (20) must be viewed 

 as representing a physical property of that substance, independent of the 

 dimensions of the mass into which it is aggregated, the constants li, q being 

 physical constants of the material. The distribution of density in a body of 

 any size is then represented by equation (25), without change in the value 

 of q, but with the maximum value of j9 chosen to agree with the total radius, 

 and with central density such as to give the surface density the fixed value 

 Pi. The distance of any particle from the center is then fixed by the angle 

 /?, which is the measure of that distance in terms of a unit of 1/g centi- 

 meters, or about 1,600 miles. 



Let rj be the radius of the nucleus already formed at a certain epoch, 

 and Tg its radius when compressed under the total load afterwards depos- 

 ited; let /, r be the central distances of any interior particle at the same 

 epochs; then r/ is to be determined as a function of r^, and / as a function 

 of r and r^; the latter function describing the history of the condensation 

 in the sense that it fixes the position of any assigned particle at any epoch, 

 if the epoch is specified by the final location of the particles which then lay 

 at the surface. A translation into time-relations could then be made for any 

 postulated law of variation in the rate of accretion. Let p^ be the central 

 density at the earlier epoch. For brevity put 



B=^ C=sin/?-/?cos/3 (36) 



to be indexed, like the various values of j9, by analogy with the values of r 

 to which they refer, as in equation (25) . 



By equation (26) the identity of the masses within the radii r/, r, at the 

 respective epochs gives the condition 



Po'C/=PoC. (37) 



similarly for the corresponding radii r', r: 



Po'C'=PoC (38) 



and from the constancy of density at the momentary surface 



P,'B:=p,B, (39) 



These equations are to determine the values of p^, /?/, /?', for any assigned 

 values of ^^ and /?, for instance first ^/ from (37) and (39) which give 



b: b. 



(40) 



