Problems relating to Underground Temperature. 371 



the ordinary diurnal maximum rates of increase and diminu- 

 tion of temperature from point to point inwards in the imme- 

 diate neighbourhood of the surface. 



In the case of Problem II. these conditions will be practically 

 fulfilled, and continue to be fulfilled, very soon after the day 

 of extraordinary temperature of which the effect is to be con- 

 sidered, and we shall have a perfectly practical solution illus- 

 trative of the consequences experienced several days or weeks 

 later at the 3-foot and 6-foot deep thermometers of the obser- 

 ving-station. The solution of Problem I., which we now pro- 

 ceed to work out, will show clearly what dimensions as to 

 space, time, and temperature may be chosen for a really prac- 

 tical illustration of its conclusions. 



Problem I., subject to the limitations we have just stated, 

 is equivalent to the following: — An infinitely small area of an 

 infinite plane terminating on one side a mass of uniform trap- 

 rock which extends up indefinitely in all directions on the other 

 side, is infinitely heated for an infinitely short time, and the 

 whole surface is instantly and for ever after maintained at a 

 constant temperature. It is required to determine the consequent 

 internal variations of temperature. 



Let the solid be doubled so as to extend to an infinite dis- 

 tance on both sides of the plane mentioned in the enunciation. 

 This plane, when no longer a boundary, we shall call the medial 

 plane. Let P, P x be two points equidistant from the medial 

 plaue in a line perpendicular to it, on each side of the portion 

 heated according to enunciation. Let a certain quantity of 

 heat Q be suddenly created in an infinitely small portion of 

 the solid round P, and at the same instant let an equal quan- 

 tity be abstracted from an infinitely small portion of the solid 

 round V . The consequent variations of temperature on the 

 two sides of the medial plane of reference will be equal and 

 opposite, being a heating effect which spreads from the medial 

 plane in one direction, and a symmetrical cooling effect spread- 

 ing from the same plane through the matter which we have 

 imagined placed on its other side. The heating effect on the 

 first side will, as is easily seen, be precisely the same as that 

 proposed for investigation in Problem L; and the thermal 

 action of the mass we have supposed added on the other side 

 will merely have the effect of maintaining the temperature of 

 the bounding plane unvaried. Now if a quantity Q of heat 

 be placed at one point (a, /3, y) of an infinite homogeneous 

 solid, the effect at any subsequent time t at any point x, y, z 

 of the solid will be expressed by the formula 



2B 2 



