OBSERVATIONS OF UNDERGROUND TEMPERATURE. 409 
lysis of imaginable secular changes of temperature, with at least thousands of mil- 
lions of years for a period. In such an application, it would be necessary to take 
into account the spherical figure of the earth as a whole. Periodic variations at 
the surface with any period less than a million* of years will, 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 in so far as they are modified by superficial 
regularities. Hence Fourier’s formule for an infinite solid, bounded on one 
side by an infinite plane, of which the temperature is made to vary arbitrarily, 
contain the proper analysis for diurnal or annual variations of terrestrial tem- 
perature, unless a theory of the effect of inequalities of surface (upon which no 
investigator 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 today. If now the annual variation of tem- 
perature were truly periodic, a complex harmonic function could be deter- 
mined 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 per- 
fectly periodic manner, since there are both great differences in the annual aver- 
age temperatures, and never-ceasing irregularities in the progress of the variation 
within each year. A full theory of the consequent variations of temperature pro- 
pagated 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 com- 
munication, 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 independent superposition of thermal 
conductions.' This principle holds rigorously in nature, except in so far as the 
* A periodic variation of external temperature of one million years’ period would give variations 
of temperature within the earth sensible to one thousand times greater depths than a similar varia- 
tion of one year’s period. Now the ordinary annual variation is reduced to »4 of its superficial 
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 ;3,). 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 zy. If the period were ten thousand million years, the variation would similarly be reduced to 
zoo 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 345 of the superficial amount) beyond the first layer of 
500 miles’ thickness. 
VOL. XXII. PART II. 5N 
