1907.] AND CONTRACTION OF THE EARTH. 213 



extending to infinity in all directions, on the supposition that at an 

 assumed initial epoch the temperature had two different constant 

 values on the two sides of a certain infinite plane ; and in finding at 

 the same time the rate of variation of temperature from point to 

 point in the solid. To these ends Fourier has demonstrated the fol- 

 lowing familiar equation for the linear conduction of heat (" Oeuvres 

 de Fourier," Tome II., p. 2J2i)' 



d% d'S 



= K 



dt d^' 



(14) 



In this equation © denotes the temperature for the depth x, at the 

 time t, so that x is the distance of any point from the middle plane, 

 and K is the conductivity of the solid rock composing the crust, 

 measured in terms of the thermal capacity of the unity of bulk ; that 

 is, K is equal to the number of units of heat which would pass across 

 one square foot of a plate of rock one foot thick in a year, when the 

 two faces of the rock are maintained at temperatures differing by i ° 

 Fahr., the unit of heat being the amount required to raise one cubic 

 foot of the rock through i ° Fahr. By careful experimental obser- 

 vations on several kinds of rock in situ Lord Kelvin found that 

 ^=3400 (cf. " Theorie Analytique de la Chaleur," Chap. Ix., § II.; 

 and especially " Le Refroidissement Seculaire du Globe Terrestre," 

 "Oeuvres de Fourier," Tome II., pp. 271-288; Thomson & Tait's 

 " Nat. Phil.," Vol. I., Part II., Appendix D ; Fisher's " Physics of 

 the Earth's Crust," second edition, p. 67 et seq.). 



If a, fi, y be any arbitary constants whatever, and f(a, (3, y) a 

 function of these quantities, and x, y, z the coordinates of any point 

 of the infinite solid, the general differential equation for the propaga- 

 tion of heat is 



d% fd'@ d'e d^e\ 



Idt^^yex'^W^^')' ^^ 



And the general integral applicable to the most varied cases, subject 

 to the appropriate surface conditions, indicated by the physical nature 

 of the problem, is 



/>+« /»+« /»+« {(„_a.)2-i-(|3_y)2-f(y_a!)2} 



e= I I 1 r*-e 4^ •f(a,^,^)dad^dri. {B) 



1/ — BO t/— ao t/— 00 



