NO. 6 AGE OF THE EARTH BECKER 21 



satisfy the partial differential equation when c, iv, and Y are constants, 

 as may easily be tested by differentiation. If c = they reduce to the 

 forms used by Kelvin. Here -y,, represents the constant temperature at 

 the surface of the cooling mass and Y the initial temperature of the cool- 

 ing mass at an infinitesimal distance from the surface. As will be proved 

 in the next paragraph, c is the constant initial temperature gradient. 



From the equations themselves it can be at once determined what 

 boundary conditions are implied. When i = 0, the upper limit of the inte- 

 gral in (2) becomes infinite, and the value of the integral itself is then 

 V7r72 ; consequently when t — and x is a positive quantity, the initial 

 distribution of temperature in tlie real solid is represented by 



v — Vq— Y -}- ex 

 while in the imaginary solid replacing outer space at the same instant 



-y — ^0= — {Y -^cx) 

 Hence equations (1) and (2) fulfil the conditions demanded by the 

 modified problem under discussion, and represent the cooling of a body in 

 which the initial temperature increased from the surface value, Y , in 

 simple proportion to the depth, x. 



The superficial temperature gradient at any time is found by making 

 .r = in ( 1 ) and is expressed liy 



\dxj^ \hnKi 

 In Kelvin's solution this quantity as well as k and Y is regarded as 

 known, while c is made equal to zero, so that (o) gives the required age. 

 In 18(32 it seemed both unobjectionable and inevitable to rely on the sur- 

 face gradient determined by observation in determining the age of the 

 earth; but it is now known that this gradient is affected by radioactivity, 

 and, therefore, that it cannot be trusted. It is the special purpose of this 

 paper to dispense with the employment of the surface gradient. This will 

 be accomplished by taking advantage of Mr. John F. Hayford's level of 

 isostatic compensation, which lies far below the level at which radio- 

 activity can affect the supply of heat. 



If appropriate values of the constants can be found, eqiiations (1) and 

 (2) can be computed for any desired age, and this computation is an easy 

 task because the value of the definite integral in (2) has been tabulated 

 by various mathematicians, the most complete table being by Mr. James 

 Burgess and printed in 1900.^ 



Kelvin employed a diffusivity, k, of 400, using the British foot and the 

 year as units. In c. g. s. units this would be 0.01178. This value was 

 obtained from experiments on the trap rock of Gallon Hill, the sand of an 



1 Trans. R. S. Edinburgh, vol. 39, 1900, p. 257. 



