410. PROFESSOR W. THOMSON ON THE REDUCTION OF 
conductivity or the specific heat of the conducting substance may vary with the 
changes of temperature to which it is subjected; and it may be accepted with 
very great confidence in the case with which we are now concerned, as it is not 
at all probable that either the conductivity or the specific heat of the rock or soil 
can vary at all sensibly under the influence of the greatest changes of tempera- 
ture experienced in their natural circumstances; and, indeed, the only cause 
we can conceive as giving rise to sensible change in these physical qualities is the 
unequal percolation of water, which we may safely assume to be confined in ordi- 
nary localities to depths of less than three feet below the surface. The particular 
mode of treatment which I propose to apply to the present subject consists in 
expressing the temperature at any depth as a complex harmonic function of the 
' time, and considering each term of this function separately, according to FourRIER’s 
formulze for the case of a simple harmonic variation of temperature, propagated 
inwards from the surface. The laws expressed by these formule may be stated 
in general terms as follows. 
11. Fourier’s Solution stated.*—If the temperature at any point of an infinite 
plane, in a solid extending infinitely in all directions, be subjected to a simple 
harmonic variation, the temperature throughout the solid on each side of this 
plane will follow everywhere according to the simple harmonic law, with epochs 
retarded equally, and with amplitudes diminished in a constant proportion for 
equal augmentations of distance. The retardation of epoch expressed in circular — 
measure (arc divided by radius) is equal to the diminution of the Napierian 
logarithm of the amplitude; and the amount of each per unit of distance is equal 

to Ni a if c denote the capacity for heat of a unit bulk of the substance, and & its 
conductivity.+ : 
12. Hence, if the complex harmonic functions expressing the varying tem- 
perature at two different depths be determined, and each term of the first be 
compared with the corresponding term of the second, the value of af TE may be 
determined either by dividing the difference of the Napierian logarithms of the © 
amplitudes or the difference of the epochs by the distance between the points. — 
The comparison of each term in the one series with the corresponding term in the 
other series gives us, therefore, two determinations of the value of di — which 
should agree perfectly, if (1) the data were perfectly accurate, if (2) the isother- ; 
mal surfaces throughout were parallel planes, and if (3) the specific heat and — 
conductivity of the soil were everywhere and always constant. a 
* For the mathematical demonstration of this solution, see Note appended to Professor EVERETI’S — 
paper, which follows the present article in the Transactions. ig 
} That is to say, the quantity of heat conducted per unit of time across a unit area of a plate of 
unit thickness, with its two surfaces permanently maintained at temperatures differing by unity, 

