208 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



Here the zone of rising temperature extends at first to a depth of a Uttle 

 over 1,000 miles, and the chief features of its history may be thus sum- 

 marized: 



Table 12. 



The rising temperatures finally cease at epoch about 24 T, with the at- 

 tainment of a maximum surface gradient of 1° in about 220 meters. 



To complete the sketch of the modified theory, in a way parallel to that 

 of Part I, there might be traced an analogous history of the nucleus during 

 the period of accretion, supposing the compressibility of the substance to 

 correspond to equation (76). But in contrast with (25) it must be noticed 

 that this could not be done by treating (74) as the general expression for 

 the density at every epoch. For by (76) the modulus of compression has 



*^^^*^'^ H=hp + h'p- (91) 



and if this is to be definite and characteristic of the substance, not depend- 

 ent on the dimensions of the mass into which it may be gathered, the 

 coefficients h, h' must be numbers depending only on the units of measure 

 used, which on account of the manner of their dependence on p^ and c can 

 be so only if the latter also are particular numbers in the same sense, 

 not arbitrary parameters depending on the size of the body. The general 

 expression for the density would have to be sought by integration of equa- 

 tion (6) with the condition (91), which would yield a family of curves cor- 

 responding to the nucleus at various stages, but of which only the final one 

 has the simple parabolic form defined by (74) . The general integral would 

 contain two constants, but one may be considered eliminated either to 



make the density finite at the center or by the condition that -ir vanish 



at the center. For this particular law of compressibility the general inte- 

 gral does not appear to be known in form sufficiently simple to make the 

 computation practicable. 



This suggests that in connection with the planetesimal hypothesis it 

 would be convenient to have a series of examples of curve-families, each of 

 which could be considered to represent the distribution of density at the 

 various stages of aggregation of a substance with a certain definite law of 

 compressibility — equations simple enough to allow the detailed computa- 

 tions, but so varied as to admit of choice in adapting to other accepted 

 data. As a help towards the discovery of such it may be asked what general 

 condition a family of density-curves must satisfy in order that a definite 

 compressibility may be deduced. 



