22 PROFESSOR J. D. EVERETT'S INVESTIGATION OF AN EXPRESSION 



soil is here supposed to be uniform as regards its thermal qualities ; and its sur- 

 face, as also the bounding planes of the stratum, horizontal. 



4. We propose to effect the integration above indicated, and deduce convenient 

 formukic for the coefficients in the expression, for the mean temperature of a 

 stratum in terms of the time. The data to be supplied by observation are the 

 temperatures at all times of the year, at two arbitrarily assumed depths. The 

 observed temperatures must be reduced to harmonic expressions (one for each 

 depth), and from these it is easy to deduce the harmonic expression for the 

 temperature at any third depth. We may therefore assume that the harmonic 

 expression for the temperature at the centre of the stratum is known ; and we shall 

 find it convenient to compare the amplitudes and epochs in this expression with 

 those in the expression for the mean temperature of the stratum. Let the 

 former expression be 



?;=Ao + A^ sin(f + Ej) + A2 sin (2f + Eg) + &c. 

 and the latter 



V = A(, + mj Aj sin {t + E^ + Uj) + m^ A^ sin {2t + Eg + a^) + &c. 

 We have to find the values of the constants lUi m.2, &c., which express the ratios 

 of the amplitudes in the two series, and of a^ a^, &.C., which expresses the differ- 

 ences of epoch. 



5. By the theory of underground conduction, the temperature at depth x 

 below the centre of the stratum will be 



i; = Ao + A,e ■' ^' . sin (t + E^^xJ'^^j 



-a-V-j- / 2'!rc \ 



+ A^e " . smlit + E^-x^-^ j +&e. 



the general term being 



+ A„e ^ . smrnf + t.„ — x J—r-l 



e denoting the base of Napierian logarithms, tt the ratio of circumference to 

 diameter, k the conductivity of the soil referred to unit volume, and c the capacity 

 for heat similarly referred. 



6. We will first suppose that the expression for the temperature at the centre 



of the stratum is simply 



V = sin t. 



Then we shall have 



v = e . sm[t-x^ljj 



and if ^ be the thickness of the stratum, the mean temperature at time t will be 



If' 



V dx 



