Februabt 7, 1908] 



SCIENCE 



229 



of the radius this law may be adequately 

 represented by a straight line, the chord of an 

 arc whose curvature is small. It would be 

 comparable with, though not identical with, 

 the superficial portion of Mr. Barus's nearly 

 rectilinear curve representing the melting 

 point of diabase as a function of depth. 

 Hence it will be sufficient to assume that the 

 initial temperature increased in simple pro- 

 portion to distance from the surface. 



It is easy to modify the Fourier equation 

 employed by Kelvin to meet this condition. 

 This equation is, strictly speaking, that of an 

 infinite solid divided by a plane, on one side 

 of which, at the initial instant, the tempera- 

 ture has one uniform value, while on the other 

 side it has another uniform value. In other 

 words, in Kelvin's problem the curvature of 

 the earth is neglected because the phenomena 

 are so superficial. 



The equation used by Kelvin of course sat- 

 isfies Fourier's law of the conduction of heat, 

 viz., 



dv d^v 



where v is temperature, t time, x distance 

 from the dividing plane and k diffusivity as- 

 sumed to be constant. It follows that 



' dx'^'' 



xfe-^'-l-iKt 



and this integrated once gives 

 dv V 



dx ^ff|^t 



E-x"-l4Kt -I- c. 



(1) 



Here V is half the difference of the two ini- 

 tial temperatures at an infinitesimal distance 

 from the dividing plane and c is a constant 

 temperature gradient. In Kelvin's solution c 

 is zero and the temperature on each side of 

 the divisional plane is uniform. A second 

 integration gives 



V llj := F • — = I 



a/tt J a 



e—^'dz -\- ex. 



(2) 



When i = {), X being positive 



V — v,= T -\- ex, 

 while for negative x 



V — Do = — y — ex. 



(3) 



This last equation represents the initial dis- 

 tribution of temperature in the hypothetical 

 solid replacing outer space in the problem of 

 a cooling earth. In these equations v„ is the 

 temperature in the dividing plane itself while 

 y is the tempeiature at an infinitesimal dis- 

 tance from the plane at the initial instant. It 

 is convenient to write v — v„^E so that E 

 is the excess of temperature of any point in 

 the solid over the temperature in the limiting 

 plane. For the present problem then 

 E=^'V + ex 



represents the initial distribution of tempera- 

 ture in the earth. 



If appropriate values of the constants can 

 be found, eqiiations (1) and (2) can be com- 

 puted for any desired age and this computa- 

 tion 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 dift'usivity, 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 Calton Hill, the sand of an 

 experimental garden and the sandstone of 

 Craigleith quarry, all at Edinburgh. Different 

 weights were given to these observations, but 

 how is not explained. Now, in considering 

 the diffusivity of the earth it does not seem 

 to me that the ragged pellicle of detrital mat- 

 ter on its surface need be considered. Over 

 large areas it is absent and in most places the 

 sedimentary rocks are saturated with water, 

 so that their own intrinsic diffusivity is a 

 minor feature of the flow of heat. The great 

 bulk of the rocks in the shell affected by cool- 

 ing are massive and at least comparable with 

 the trap of Calton Hill, which is chiefly com- 

 posed of Carboniferous basalt and andesite. 

 The conductivity of this rock was observed by 

 Forbes and Thomson (Kelvin) for no less 

 than eighteen years, the thermal capacity was 

 determined by Eegnault, so that the value of 

 the diffusivity, 0.00786, is undoubtedly very 

 accurate. It does not stand alone. A com- 



' Trans. B. 8. Edinburgh, Vol. 39, 1900, p. 257. 



