NORMAL COOLING OF THE GLOBE. 



131 



a mantle of ice. Under these circumstances the temperature at points in the crust 

 will fall towards that at the surface in a way defined thus : 

 Let 



u a == the initial excess of the temperature of the crust over the constant tempera- 

 ture at the surface, 

 M = the temperature of the crust at a depth x at any time t after the initial 



epoch, 

 « 2 =:the diffusivity of the crust = 400, about (Thomson*), for foot and year as 



units, 

 tt = 3.14154-, 



.- = subject- variable of integration, 

 e = Naperian base. 

 Then the fall of temperature u — u is expressed by the equation 



x 



iay't 

 u — u = u a ( 1 — - J e 2 </ z \ 



The following table gives the values of _5 for various values of the depth x 



and the time t: 



Values of Ratio 



u — u 



for Different Times and Depths. 



To illustrate the application of the table, suppose the mean annual temperature 

 over the Alaskan region to have been 10° F. (the present mean annual temperature 

 of northern Alaska) since the initial epoch, and suppose that the temperature of the 

 crust was initially 60° F. Then the u Q of the formula is 50°. At the end of a year 

 from the initial epoch the temperature at a depth of 40 feet (see table) would fall 

 0.157 X 50°, or about 8° ; i. e., the temperature at that depth would be 60° — 8° := 52°. 

 At the end of 25 years the temperature, at the depth of 40 feet, would fall to about 

 60° — 0.777 X 50° = 21°; but the fall would be hardly perceptible at a depth of 400 

 feet, etc. 



: See Treatise on Natural Philosophy, by Thomson and Tait, Vol. 1, Part u, Appendix I'. 

 XVIII— Bull. Geol. Soc. Am., Vol. 1, 1889. 



