374 Problems relating to Underground Temperature. 



the fire is applied and removed) the temperature increases to a 

 maximum at a certain time, and then diminishes to zero again 



Fig. 1. 



after an infinite time ; the ultimate law of diminution being in- 

 versely as the square root of the fifth power of the time. The 

 time when the maximum temperature is acquired at a distance 



r 2 

 r from the place where the fire was applied, is y^y, or, accord- 

 ing to the value we have found for the trap-rock, 9 ». _ of a 



year. Thus it appears that at one French foot from the place 

 of the fire, the maximum temperature is acquired a day and 

 a half (more exactly 1*54 day) after the application and 

 removal of the fire. At 15*4 French feet from the fire the 

 maximum temperature is reached just a year from the begin- 

 ning; and at 1540 feet the maximum is reached in 10,000 

 years. The law of variation of temperature with time is shown 

 by the curve of fig. 2, the ordinates of which represent tem- 

 peratures, and the abscissas times. 



Fig. 2. 



From these results we can readily see how the circumstances 

 of the proposed problem may be actually realized, if not rigo- 

 rously, yet to any desired degree of approximation, in the 

 manner supposed — namely, by keeping a fire burning for a cer- 

 tain time over a small area of rock, and then removing it and 

 cooling the surface. 



