Observations of Underground Temperature. 29 



the specific heat of the conducting substance may vary with the 

 changes of temperature to which it is subjected ; and it may be 

 accepted with very great confidence in the case with which we 

 are now concerned, as it is not at all probable that either the 

 conductivity or the specific heat of the rock or soil can vary at all 

 sensibly under the influence of the greatest changes of tempera- 

 ture experienced in their natural circumstances ; and, indeed, the 

 only cause we can conceive as giving rise to sensible change in 

 these physical qualities is the unequal percolation of water, which 

 we may safely assume to be confined in ordinary localities to 

 depths of less than three feet below the surface. The particular 

 mode of treatment which I propose to apply to the present- sub- 

 ject consists in expressing the temperature at any depth as a 

 complex harmonic function of the time, and considering each 

 term of this function separately, according to Fourier's formula? 

 for the case of a simple harmonic variation of temperature, pro- 

 pagated inwards from the surface. The laws expressed by these 

 formulae may be stated in general terms as follows. 



11. Fourier's Solution stated*. — If the temperature at any 

 point of an infinite plane, in a solid extending infinitely in all 

 directions, be subjected to a simple harmonic variation, the tem- 

 perature throughout the solid on each side of this plane will 

 follow everywhere according to the simple harmonic law, with 

 epochs retarded equally, and with amplitudes diminished in a 

 constant proportion for equal augmentations of distance. The 

 retardation of epoch expressed in circular measure (arc divided 

 by radius) is equal to the diminution of the Napierian logarithm 

 of the amplitude ; and the amount of each per unit of distance 



is equal to a / stt, , if c denote the capacity for heat of a unit bulk 



of the substance, and k its conductivity f. 



12. Hence, if the complex harmonic functions expressing the 

 varying temperature at two different depths be determined, and 

 each term of the first be compared with the corresponding term 



of the second, the value of a / ~7 may be determined either by 



dividing the difference of the Napierian logarithms of the ampli- 

 tudes, or the difference of the epochs by the distance between the 

 points. The comparison of each term in the one series with the 



* For the mathematical demonstration of this solution, see Note ap- 

 pended to Professor Everett's paper, which follows the present article in 

 the Transactions. 



f That is to say, the quantity of heat conducted per unit of time across 

 a unit area of a plate of unit thickness, with its two surfaces permanently 

 maintained at temperatures differing by unity. 



