406 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



The change in a given interval of time of the amount of energy 

 contained in the ground beneath a unit of surface is equal to the 

 change in the average temperature of the ground from the surface 

 down to the depth at which the variations are inappreciable, 

 multiplied by the water value of a prism cut from this ground 

 from the unit surface down to the same depth. 



Hence the energy stored in the earth attains its extreme values 

 simultaneously with those of the average temperature of the ground 

 if in determining these latter we consider the temperatures down 

 to the depths at which the variations become inappreciable. 



The equation (23) allows of a t very simple geometrical presen- 

 tation (see fig. 56). 



Let the depths h be shown as ordinates counted positively down- 

 ward, and the temperatures 6 as abscissae, then will the distribu- 

 tion of temperature in the ground at the moment t and down to 

 the depth h be represented by the curve A t B x as in fig. 56. 



If now the distribution of tem- 

 perature is different at the mo- 

 ment t 2 and if it be represented by 

 the curve ^4 2 J3 2 then the area A x 

 B l B 2 A 2 which is included between 

 the two curves and the axis of 

 abscissas and the line parallel to 

 FIG - 5 6 this at the depth h, and which 



may be designated by /, becomes at once a measure of the added 

 quantity of heat, since 



*/ 



f - J (0 2 - e x ) dh 



At the same time this method of presentation gives immediately 

 information as to the direction in which the movement of the heat 

 is taking place in the different strata of the ground at the given 

 times, since the course of the lines allows us to recognize immediately 

 whether the temperatures increase or diminish downward. The 

 direction of the flow of heat is shown by arrows in fig. 56. 



Because of the great advantages offered by the consideration of 

 this curve, I will give it a special name, the "tautochrone, " since 

 each such curve presents the temperatures that prevail at the 

 given time at the different depths. 



Incidentally it may be remarked that from these curves we can 

 construct a remarkably instructive picture. 



