578 Mr. H. Jeffreys on the Compression 



to be available on the contraction hypothesis appears to be 

 quite adequate. 



II. The effect on underground temperatures of a uniform 

 distribution of radioactive matter through a horizontal layer. 



Let the number of calories generated per unit time per 

 cubic centimetre be A. 



Let the temperature at depth x at time t be V. 



Let the depth of the radioactive layer be X. 



Then, when x is between and X, V satisfies the equation 



BV_ 72 ^V_A 



-dt n ^ 2 ~ c/ )' w 



and when x is greater than X, we have 



Here p is the density of the rocks and c their specific 

 heat. If k is their conductivity, 



h 2 = k/cp (3) 



The boundary conditions are that when £ = 0, Y = mx + S 

 for all values of the depth, and when x = 0, V = for all 

 values of the time. Further, V and ~dYfox must both be 

 continuous at a?=X. 



Evidently V= — A (a?— X) 2 /2& satisfies the first equation. 

 Substitute then 



Y = u + A{X 2 - (x—X) 2 }/2k when x is less than X, . (4) 

 and 



V = u + A\ 2 /2k when x is greater than X. (5) 



Then 'du 12 d 2 w , 



=r nr^—g =0 everywhere (6) 



^t B^' 



u = when # = ; u and 'du/'dx are both continuous at x—\. 

 When t = 0, 



u = mx + $ — A{\ 2 — (a-\) 2 }/2k when x is less than X.] 



n — mx f S — AX 2 /2/c when x is greater than X.J 



(7) 



Now it has been shown by Fourier * that the solution 

 of (6) that makes u—f[x) when £ = 0, the value of f(x) being 

 specified for all values of x from — oo to + oo , is 



'^_Jqe- q y\x + 2qh x /7) (8) 



* ' Analytical Theory of Heat/ p. 354. 



