ON UNDERGROUND TEMPERATURE. 251 



" The annual wave of temperature is propagated downward from tlie 

 surface, at a rate which depends on the nature of the soil, and is on the 

 average rather greater than a foot per week ; while at the same time the 

 amplitude (or magnitude) of the wave diminishes in a ratio also de- 

 pendent on the soil, and amounting on the average to a halving of the 

 amplitude for every five or six feet of descent. 



" Supposing the soil to be uniform, the surface to be plane, and the 

 propagation of heat to be effected solely by conduction, a simple har- 

 monic variation of temperature at the surface (which we may call in 

 popular language a simple wave of temperature) will be propagated 

 downward with a uniform velocity, and with amplitude diminishing in 

 geometrical progression. There will, moreover, be a defiuite relation 

 between the ratio of this progression and the velocity of propagation, 

 so that if the one is given the other can be computed. In fact, we shall 

 have — 



f) _ time of i)ropagation from one depth to another 

 period of variation 



amplitude at 1st depth 



= Napierian logarithm of 



amplitude at 2d depth 



difference of d epths frt o 

 -v/period of vi 



depths J7tc 

 variation^ ^^ 



where - denotes 3.1416; c, thermal capacity per unit-voluuie; and. A, con- 

 ductivity.* 



" If the variation of temperature at the surface, instead of being simple 

 harmonic, be any periodic variation whatever, it can be reduced by 

 Fourier's method to the sum of a number of simple harmonic variations, 

 and each of these variations will be propagated according to the above 

 law, unaffected by the rest. 



" As the square root of the number of days in the year is almost exactly 

 19, the above formula shows that the annual wave is propagated 19 times 

 as fast as the diurnal wave, and that the falling off in amplitude is tlie 

 same in one foot for the diurnal wave as in 19 feet for the annual wave. 



" Of the different simple harmonic components which make up the 

 whole variation at the surface, those of longest period are propagated 

 downward most quickly, and die away most slowly. For this reason, 



*The numerical value of the co-efficient^ /-_? as given in Professor Everett's Discus- 



^'-i-" 



siou of the Observations at tbe Greenwich Observatory, 1860, is as follows: 



From the Greenwich observations 0.0918 



From Caltou Hill, trap-rock 0.115t> 



From Experimental Garden, sand 0.1098 



From Craigleith Quarry, sandstone 0.0674 



fhe diminution of /!^ indicates either a decrease in capacity for heat or an increase 

 in conductivity. 



