214 GEOPHYSICAL THEORY UNDER THE PLA.NETESIMAL HYPOTHESIS. 



or as power-series 



Since fx is in effect a physical constant of the material, equation (121) 

 shows that any particular value of u will determine a spherical surface, 

 of varying radius according to the varying value of p^, but always passing 

 through the same material particles, so as to inclose the mass determined 

 by that equation. If, then, u, v be two values of u corresponding to two 

 definite particles, a', /?' and a", /J", their distances from the center at two 

 stages of the compression when the central densities are p,/, p^' respectively, 

 then the definition of u gives 



u =p,'ipia' =^o"M«" V =/>o'Mi5' =p,"ipi^' 



whence 



showing that the distances of two particles from the center remain in the 

 same ratio as the condensation progresses. This is precisely the kind of 

 contraction by which distortion of elements is avoided. 



The constant /j. would be determined by the mean and surface densities 

 at a given epoch from the equations 



p,=poco{u0 Pm=^Po—^ Ui=Po^tir^ (124) 



t*, 



which would yield also the central density p^ at that epoch; then for any 

 epoch the first of these would determine the central density for any assigned 

 total radius. 



The determination of the exact distribution of density under these con- 

 ditions would rest on the computation of the coefficients so that R satisfies 

 the differential equation (118); but a sufficient idea of the curve may be 

 gathered simply by comparison of formula (117) with (96) and (20). All of 



these belong to the class (104), with exponents :^, -^, and 1 respectively, so 



that the first is a sort of average of the other two. With the same assigned 

 mean and surface densities, it may be inferred that the hypothesis now 

 considered would result in densities less in the outer strata and greater in 

 the central portion than those given by the Laplacian formula (25), but 

 deviating in that way to a less extent than those computed from (94). 

 Such departures are opposite to the corrections which the Laplacian law 

 seems to require, though the data from observation are but meager; so 

 that (117) must probably be rejected, as not giving a satisfactory approx- 

 imation to the actual distribution of density within the earth. The only 

 satisfactory curves of class (104) would seem to be those with exponents 

 somewhat larger than unity. 



The intermediate character of (117) as compared with (20) and (96) 

 appears also in the phenomena which would attend unlimited accretion. 



