404 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



they are from the meteorological point of view, still the material 

 for reply offered by a superabundance of observations of earth tem- 

 peratures is extremely scarce, since in only very few cases has the 

 volume capacity of the appropriate earth been directly determined 

 and therefore the essential datum is missing. 



The above questions will now be first answered theoretically and 

 then an attempt be made to see how far the formula can be con- 

 verted into numbers; also for simplicity it will first be assumed 

 either that the temperatures remain always above the freezing 

 point or that the earth is wholly free from water. 



This being assumed the first of the two questions, i. e., the increase 

 of energy contained in the ground within a given interval of time 

 t 2 is ansAvered by the following consideration: 



Let C be the thermal capacity of the unit volume, h the distance 

 of any point from the surface of the earth, reckoned positive down- 

 ward,^ the temperature of the earth at this point at the moment t v 

 2 the corresponding temperature at the moment t 2 ; imagine a 

 prism cut from the ground beneath the unit surface, then an infi- 

 nitely thin horizontal element of this prism having the thickness dh 

 receives in the given interval of time the quantity of heat repre- 

 sented by 



C (0 2 - 0,) dh 



The quantity of heat received by the whole prism to the depth H, 

 that is to say, the change of the energy within the prism results 

 from the equation 



H 



u 2 -ih = j C (d 2 - 0J dh 







or if C is constant 



H 



u 2 -u t = CJid.-dJ dh (23) 







In this equation 6 t and 6 2 are functions of h such that with increas- 

 ing values of h they very rapidly approach toward equality, so that 

 if great accuracy be not required the difference 2 — 6 t may be 

 assumed to be o when H = 10 meters even when t t and t 2 differ 

 greatly. If we consider only a short interval of time, such as 24 

 hours, then we may assume that this limit is reached when H — 1 

 and can put X = d 2 for this depth. 



