obtained by putting x 2 +y 2 + z 2 = r 2 and x = r cos 6. We thus 

 have 



4kt, 



to Underground Temperature. 373 



y 2 + z 2 = r 2 and a = rc 



T (Atf) 2 cos 6 . re &* 



107T2 



which expresses the temperature assumed at a time t after the 

 application of the fire, by a point of the solid at a distance r 

 from the point of the surface where the fire was applied, and 

 situated in a direction inclined at an angle 6 to the vertical 

 through this point. From this expression we conclude: — 



(1) The simultaneous temperatures at different points equi- 

 distant from the position of the fire are simply proportional to 

 the distances of these points from the plane surface. 



(2) The law of variation of temperature with distance in 

 any one line from the place where the fire was applied is the 

 same at all times. 



(3) The law of variation of temperature with time is the 

 same at all points of the solid. 



(4) Corresponding distances in the law of variation with 

 distance increase in proportion to the square root of the time 

 from the application and removal of the fire ; and therefore, 

 of course, corresponding times in the law of variation with 

 time are proportional to the squares of the distances. 



(5) The maximum value of the temperature, in the law of 

 variation with distance, diminishes inversely as the square of 

 the increasing time. 



(6) The maximum value of the temperature in the law of 

 variation with time, at any one point of the rock, is inversely 

 as the fourth power of the distance from the place where the 

 fire was applied. 



(7) At any one time subsequent to the application of the 

 fire, the temperature increases in any direction from the place 

 where the fire was applied to a maximum at a distance equal 

 to V 2kt, and beyond that falls to zero at an infinite distance 

 in every direction. The value of k for the trap-rock of Calton 

 Hill being 141, when a year is taken as the unit of time and 

 a British foot the unit of space, the radius of the hemispherical 

 surface of maximum temperature is therefore 16*8 x s/t feet. 

 Thus at the end of one year it is 16*8 feet, at the end of 

 10,000 years it is 1689 feet, from the origin. The curve of 

 fig. 1 shows graphically the law of variation of temperature 

 with distance. The ordinates of the curve are proportional to 

 the temperatures, and the corresponding abscissas to the dis- 

 tances from the origin or place of application of the fire. 



(8) At any one point at a finite distance within the solid 

 (which, by hypothesis, is at temperature zero at the instant when 



