222 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



mass everywhere shrinks toward the center, even while the temperature is 

 rising in the outer strata. In particular the surface falls at a rate propor- 

 tional to the temperature-gradient there, so that its shrinkage is first accel- 

 erated and later retarded, the most rapid fall occurring when the zone of 

 rising temperatures is disappearing. 

 In this case equations (131) become 



aXdd d /.dd\ 



r dr or 



«t) 



The former is everywhere negative; but the latter, if X is constant, is nega- 

 tive where the temperature-curve is convex upward and positive where it 

 is concave; moreover, where £2 is negative it is numerically less than e,. 

 Thus the adjustment of density to temperature demands, in the interior, 

 both horizontal and vertical contraction, with the former more marked; in 

 the superficial strata, as long as the temperature rises, horizontal contrac- 

 tion and vertical expansion. To the extent to which the mass resists defor- 

 mation there are therefore developed at all depths a horizontal thrust and 

 a vertical tension, which accumulate at rate proportional to £2~^i! ^^^^ 

 may be written . . ^n 



e,-., = aAr-(--) 



Numerical or graphic differentiation from the tabulated temperatures shows 

 that at any one epoch the shearing stress thus indicated is roughly propor- 

 tional to the square of the distance from the center. 



The maximum possible shortening of the total radius, corresponding to 

 a reduction of the temperature from its initial value 6 down to zero through- 

 out the mass is 



''Jo 



Jr,= — r / adr^dr 



which reduces to 



■'"^"^7, 



Ar^ = -^ I epr^ dr 



AnJr^^ 



The value of ^r, between any two epochs can be found conveniently by 

 comparison of this last integral with that coming from (128) integrated 

 with respect to i by substitution of the appropriate value of dd. The 



computation shows that this total shortening, with — constant, would be 



something less than 10 per cent of the whole radius; and graphic integra- 

 tion from table 6 shows that about one-third of this would be accomplished 

 during the epoch of rising temperatures, so that up to the time of maximum 

 surface-gradient the circumference would diminish by about 800 miles. 

 This number would, however, vary considerably under the different hypoth- 

 eses, of which the one adopted for the moment is such as rather to exag- 

 gerate the influence of the shrinkage in the central parts, and unless the 

 conductivity has been seriously underrated must probably be held to refer 



