343 
or in an infinite solid experiencing at every point of an infinite 
plane through it a variation of temperature according to the same 
elementary law. In any locality in which the surface of the earth 
is sensibly plane and uniform all round to distances amounting at 
least to considerable multiples of the depth of the lowest thermo- 
meter, and in which the conducting power of the soil or rock below 
the surface is perfectly uniform to like distances round and below 
the thermometers, this theory must necessarily be found in exces- 
sively close agreement with the observed results. The comparison 
which is made in the investigations now brought forward must be 
regarded, therefore, not as a test of the correctness of a theory 
which has mathematical certainty, but as a means of finding how 
much the law of propagation of heat into the soil is affected by the 
very notable deviations from the assumed conditions of uniformity 
as to surface, or by possible inequalities of underground conductivity 
existing in the localities of observation. When those conditions of 
uniformity are perfectly fulfilled both by the surface and by the 
substance below it, the law of variation in the interior produced by 
a simple harmonic variation of temperature at the surface, as in- 
vestigated by Fourier, may be stated in general terms in the three 
following propositions : — (1.) The temperature at every interior point 
varies according to the simple harmonic law, in a period retarded by 
an equal interval of time, and with an amplitude diminished in one 
and the same proportion, for all equal additions of depth. (2.) The 
absolute measure in ratio of arc to radius, for the retardation of 
phase, is equal to the diminution of the Napierian logarithm of the 
amplitude; and each of these, reckoned per unit of length as to 
augmentation of distance from the surface, is equal to the square 
root of the quotient obtained by dividing the product of the ratio of 
the circumference of a circle to its diameter into the thermal capa- 
city of a unit of bulk of the solid, by the thermal conductivity of the 
same estimated for the period of the variation as unity of time. (3 ) 
For different periods, the retardations of phase, measured each in 
terms of a whole period, and the diminutions of the logarithm of the 
amplitude, all reckoned per unit of depth, are inversely proportional 
to the square roots of the periods. 
The first series of observations examined by the method thus de- 
scribed were those instituted by Professor Forbes, and conducted 
under his superintendence during five years, in three localities of 
