THERMODYNAMIC THEORY. 229 



d-d,=—,{v,-vy + ^^^{v,-v) (165) 



the former corresponding exactly to (22) above. 



In the equation (165) for the temperature the first term corresponds, 

 except as to the value of the coefficient, to that which alone occurred in 

 the analogous equation in Part I for the case of constant specific heat; 

 while the second term is new and may be considered as the corrective term 

 needed to take account of the coefficient of expansion, which occurs as one 

 of its factors. The numerical value however of this corrective term is 

 small; for instance, with iCi = 4XlO", ^i = 300 (the absolute temperature 



of the surface), ai = 2XlO"~^ and a = -^J, its maximum value, which occurs 



at the center where the density is greatest, is about 80°, while it contributes 

 an initial surface gradient of 1° in 14,000 meters. The value here used for 

 Ki comes by comparison with the value deduced for H^ in Part I, which, as 

 is seen by (158), exceeds it by only one four-hundredth part and is known 

 to range well with the results of direct measurement on surface rock. 



It appears thus that to take account of all the measured mechanical- 

 thermal constants relating to surface-rock it is sufficient to include only 

 this trivial variation from the distribution of temperature specified by the 

 first term of (165), which in algebraic form is identical with that occurring 

 in the corresponding case in Part I. But the absolute value of this prin- 

 cipal term depends on the coefficient A, which is here left undetermined, 

 instead of having the definite value hj2a as under the former hypothesis. 

 So far as concerns consistency with the conditions thus far assumed, this 

 constant might be chosen so as to give any assigned value to the central 

 temperature for example, or to the surface-gradient at some particular 

 epoch of the conduction. But to make even the maximum surface-gradient 

 match the present observed value of 1° in 30 meters it would be necessary 

 to choose A so that the initial temperatures, while similar in relative mag- 

 nitude at the various depths, would be in absolute value several times as 

 large as those listed in column 8 of table 2. This would perhaps be a rather 

 extreme supposition, though there seem to be no definite data to the contrary. 



A suggestion from another source as to a fair mean value to be taken 

 for A is the following: 



In the assumed expression (141) there are the two terms: ad, which may 

 be considered for vividness as the kinetic portion, and (f{v) the potential 

 portion, of the intrinsic energy e. The latter part, depending conceivably 

 simply upon the mutual distances of the constituent particles, may by 

 analogy with the dynamics of particles plausibly be supposed always to 

 increase as v increases. Now from (149), transformed by substitution of 

 the constants as determined in terms of the surface-data, comes 



,'(»)=(.^-^-<)(^-i,) + /f,„A (166) 



