416 PROFESSOR W. THOMSON ON THE REDUCTION OF 
The values which were found for A, should represent the annual mean tempera- 
tures. They differ slightly from the annual means shown in the Royal Observatory 
Report, which, derived 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 paper, their consideration must be deferred to a future occasion. 
18. Theoretical Discusston.—The mean value of the coefficients in the last 
line of the table, being obtained from so considerable a number of years, can be — 
but very little influenced by irregularities from year to year, and must therefore 
correspond to harmonic 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 varia- 
tion 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 nonperiodic variations of internal temperature so expressed. - 
20. Let us, accordingly, now consider the complex harmonic functions corre- 
sponding 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 represents the resultant simple harmonic variation of the period to — 
which it corresponds, in the manner shown by the proposition and formule of 
§ 3 above. | 
21. On looking to the annual and semi-annual 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 
amplitudes, and the rate of retardation of the epoch between the points of obser- 
vation in order of depth :— 

