Observations of Underground Temperature. 123 



The values which were found for A should represent the 

 annual mean temperatures. They differ slightly from the an- 

 nual means shown in the Royal Observatory Report, which, de- 

 rived as they are from a direct summation of all the weekly 

 observations, must be more accurate. The variations, and the 

 final average values of these annual means, present topics for 

 investigation of the highest interest and importance, as I have 

 remarked elsewhere (see British Association's Report, Section A, 

 Glasgow, 1855) • but as they do not belong to the special sub- 

 ject of the present papei', their consideration must be deferred 

 to a future occasion. 



18. Theoretical Discussion. — The mean value of the coefficients 

 in the last Hue of the Table being obtained from so considerable 

 a number of years, can be but very little influenced by irregu- 

 larities from year to year, and must therefore correspond to har- 

 monic functions for the different depths, which would express 

 truly periodic variations of internal temperature consequent upon 

 a continued periodical variation of temperature at the surface. 



19. According to the principle of the superposition of thermal 

 conductions, the difference between this continuous harmonic 

 function of five terms for any one of the depths, and the actual 

 temperature there at the corresponding time of each year, would 

 be the real temperature consequent upon a certain real variation 

 of superficial temperature. Hence the coefficients shown in the 

 preceding Table afford the data, first by their mean values, to 

 test the theory explained above for simple harmonic variations, 

 and to estimate the conductivity of the soil or rock, as I propose 

 now to do ; and secondly, as I may attempt on a future occasion, 

 to express analytically the residual variations which depend on 

 the inequalities of climate from year to year, and to apply the 

 mathematical theory of conduction to the non-periodic variations 

 of internal temperature so expressed. 



20. Let us accordingly now consider the complex harmonic 

 functions corresponding to the mean coefficients of the preceding- 

 Table ; and in the first place, let us reduce the double harmonic 

 series in each case to series in each of which a single term repre- 

 sents the resultant simple harmonic variation of the period to 

 which it corresponds, in the manner shown by the proposition 

 and formula? of § 3 above. 



21. On looking to the annual and semiannual terms of the 

 series so found, we see that their amplitudes diminish, and their 

 epochs of maximum augment, with considerable regularity from 

 the less to the greater depths. The following Table shows, for 

 the annual terms, the logarithmic rate of diminution of the am- 

 plitudes, and the rate of retardation of the epoch between the 

 points of observation in order of depth : — 



