128 Prof. W. Thomson on the Reduction of 



pancies are owing to variations of k and c not in inverse propor- 

 tion one to the other, I have taken Fourier's equation 



dv _ , d?v dk do 

 dt dx 1 dx dx' 



where v denotes the temperature at time t, and at a distance x 

 from an isothermal plane of reference (a horizontal plane through 

 thermometer No. 1., for instance) ; k the conductivity, varying 

 with x; and c the capacity for heat of a unit of volume, which 

 may also vary with x. In this equation I have taken 



(^-q), 



where P and Q are functions of x, assumed so as to express, as 

 nearly as may be, the logarithmic amplitudes, and the epochs, 

 deduced from observation. I have thus obtained two equations 

 of condition, from which I have determined k and c, as functions 

 of x. The problem of finding what must be the conductivity 

 and the specific heat at different depths below the surface, in 

 order that, with all the other conditions of uniformity perfectly 

 fulfilled, the annual harmonic variation may be exactly that which 

 we have found on the average of the eighteen years' term at 

 Calton Hill, is thus solved. The result is, however, far from 

 satisfactory. The small variations in the values of P and Q 

 which we have found in the representation of the observed tem- 

 peratures require very large and seemingly unnatural variations 

 in the values of k and c. 



30. I can only infer that the residual discrepancies from Fou- 

 rier's formula shown in Table VIII. are not with any probability 

 attributable to variations of conductivity and specific heat in the 

 rock, and conclude that they are to be explained by irregularities, 

 physical and formal, in the surface. It is possible, indeed, that 

 thermometric errors may have considerable influence, since there 

 is necessarily some uncertainty in the corrections estimated for 

 the temperatures of the different portions of the columns of liquid 

 above the bulbs; and before putting much confidence in the dis- 

 crepancies we have found as true expressions of the deviations 

 in the natural circumstances from Fourier's conditions, a careful 

 estimate of the probable or possible amount of error in the ob- 

 served temperatures should be made. That even with perfect 

 data of observation as great discrepancies should still be found 

 in final reductions such as we have made, need not be unexpected 

 when we consider the nature of the locality, which is described 

 by Professor Forbes in the following terms : — 



The position chosen for placing the thermometer was below 

 the surface "in the Observatory enclosure on the Calton Hill, at 



