and the Earth's Thermal History. 109 



The first two terras of equation 10 are those which would obtain 

 if no radio-thermal energy were present l (i.e. x = and A = 0). 

 They, therefore, represent the contribution to the temperature 

 gradient furnished by the eai'th's original thermal condition, and the 

 third term represents that portion of the gradient which is due to 

 radio-activity. The total heat q which issues from unit area of the 

 earth's surface is obtained by multiplying equation 10 throughout by 

 k, since q = k d0/0x. "We may therefore write — 

 Total heat flow per unit area due to initial thermal state 



= ml + Slc/hy/^i. 

 Total heat flow per unit area due to radio-activity, R, 



a L ahyj Trt J 



or, R = q- [mjc + Slc/h y/~f] (11) 



In Part I (p. 64) it was assumed that R = q = A/a, which would 

 be strictly true only after an infinite time. Here it is assumed that 

 \/n of the earth's heat is due to radio-activity, i.e. R = qjn. Hence 

 from equation 1 1 we have 



qjn = q - [mh + Skjhy/^i] 

 or, since q = k dO/dx 

 . {dO/dx)jn — de/dx — \_m + S/hy/^t]- 

 Here the only unknown is n, which is given bv 



n = '^ (12) 



dQJdx — m — iS/hy/nt 



= 4/3. 



This result could also be obtained as follows : without radio-activity 



the temperature gradient at the surface after 1,600 million years of 



cooling would be — 



dOjdx = m + ojh \l irt 



= 0-00005 + 0-00003 



= 0-00008° C. per cm. 

 But the actual gradient is 0-00032° C, so that three-fourths of the 

 total present heat flow is due to radio-activity. 



The above discussion clearly proves that the earth could have 

 cooled from a state in which it was molten at the surface to its 

 present condition (as implied by the temperature gradient at the 

 surface) even if three-fourths of the heat flow be due to radio- 

 activity. This interesting conclusion must, however, be tested 

 further by reference to volcanic temperatures. If the assumptions 

 made bear a close relation to the actual conditions, they must lead to 

 temperatures in depth in keeping with the facts of vulcanism. 



11. Beaking on the Age of the Earth. 

 It is interesting to notice that in the .absence of radio-thermal 

 energy the age of the earth given by the data used in this paper 

 would be twenty-two million years, the equation 



being derived from the non-radio-active part of equation 10. 

 1 See equation 61, Ingersoll & Zobel, loc. cit., p. 89. 



