of the Earth's Crust in Cooling. 581 



for A in the previous solution and integrate with regard to 

 \ from zero to infinity. 



In particular, in the case of the exponential law used by 

 Holmes*, </>(c© ) — <£(V) must be equal to Ae~ ak ; in this 

 case (/>'(A,) = Aae~ aX } and on substituting in V and integrating 

 we find 



v =^ + ( s - < A) Er£ 2 i7I + A( 1 — ->• <"> 



It must be noted that equation (15) is only valid when 

 t is so great that \/2h& is small, and similarly, (16) only 

 holds when l/2aht* is small. On the other hand, u is an 

 even function of X, and therefore contains no term in X 3 . 

 Hence (15) and (16) will hold provided the fourth power of 

 X can be neglected. 



Differentiating (16) with regard to x and then putting <r 

 zero, we find that the known velocity gradient at the surface 

 gives a quadratic equation to determine a. The positive 

 root of this equation is a = 4'14x 10~ 7 /1 cm. 



IV. The straining of the crust in cooling. 



If it be assumed that throughout the process of cooling the 

 earth preserves a state of spherical symmetry, then it is 

 evident that as the changes of temperature are not the same 

 at all points, a state of strain must be set up, consequent on 

 the variations of volume that take place. 



Thus, consider a shell of internal radius r and external 

 radius r + Sr. 



Let the coefficient of linear expansion with temperature 

 be n. It is supposed to be a function of the temperature. 



Let the initial density of the shell be p. 



Also let the rise in temperature in a definite time be V, 

 which is supposed small. Then V is a function of r. The 

 density of the shell will change at the same time from p to 

 p{l — 3nV). Let the radius at the same time change to 

 r(l + «). 



The external radius will change to 



r(l + «) + 8r{l+i;(«0}. 



Hence the mass of the shell after the change of temperature is 

 ±iri*p8r j 1 + 2cc + ^- (m) - 3nV j . 



* Geol. Mag. March 1915, p. 108. 



