28 Prof. W. Thomson on the Reduction of 



at points below the surface, give rise to variations of temperature 

 not appreciably influenced by the general curvature, and sensibly 

 agreeing with what would be produced if the surface were an 

 infinite plane, except insofar as they are modified by superficial 

 irregularities. Hence Fourier's formula? for an infinite solid, 

 bounded on one side by an infinite plane, of which the tempera- 

 ture is made to vary arbitrarily, contain the proper analysis for 

 diurnal or annual variations of terrestrial temperature, unless a 

 theory of the effect of inequalities of surface (upon which no in- 

 vestigator has yet ventured) is aimed at. 



9. The effect of diurnal variations of temperature becomes 

 insensible at so small a distance below the surface, that in most 

 localities irregularities of soil and drainage must prevent any very 

 satisfactory theoretical treatment of their inward progression and 

 extinction from being carried out. At depths exceeding three 

 feet below the surface, all periodic effects of daily variations of 

 temperature become insensible in most soils, and the observable 

 changes are those due to a daily average, varying from day to 

 day. If now the annual variation of temperature were truly 

 periodic, a complex harmonic function could be determined to 

 represent for all time the temperature at three feet or any greater 

 depth. But in reality the annual variation is very far from 

 recurring in a perfectly periodic manner, since there are both 

 great differences in the annual average temperatures, and never- 

 ceasing irregularities in the progress of the variation within each 

 year. A full theory of the consequent variations of temperature 

 propagated downwards, must include the consideration of 

 non-periodic changes; but the most convenient first step is 

 that which I propose to take in the present communication, in 

 which the average annual variations for groups of years will be 

 discussed according to the laws to which periodic variations are 

 subject. 



10. The method which Fourier has given for treating this and 

 other similar problems is founded on the principle of the inde- 

 pendent superposition of thermal conductions. This principle 

 holds rigorously in nature, except insofar as the conductivity or 



amount at a depth of 25 French feet, and is scarcely sensible at a depth of 

 50 French feet (being there reduced, in such rock as that of Calton Hill, 

 to x^o). Hence, at a depth of 50,000 French feet, or about ten English 

 miles, a variation having one million years for its period would be reduced 

 to z fo . If the period were ten thousand million years, the variation would 

 similarly be reduced to T ^^ at 1000 miles' depth, and would be to some 

 appreciable extent affected by the spherical figure of the whole earth, 

 although to only a very small extent, since there would be comparatively but 

 very little change of temperature (less than J ff of the superficial amount) 

 beyond the first layer of 500 miles' thickness. 



