OBSERVATIONS OF UNDERGROUND TEMPERATURE. 421 
variations of # and ¢ not in inverse proportion one to the other, I have taken 
FouriEr’s equation 
du pe i dk du 
dt da” dx dx’ 

where v denotes the temperature at time ¢, and at a distance x from an isothermal 
plane of reference (a horizontal plane through thermometer No. I., for instance) ; 
& the conductivity, varying with 2; and ¢ the capacity for heat of a unit of 
volume, which may also vary with x. In this equation I have taken 
v=ae” cos (ir -@) 5 
where P and Q are functions of w, 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 / and «¢, 
as functions of z. 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 repre- 
sentation of the observed temperatures, require very large and seemingly un- 
natural variations in the values of & and c. . 
30. I can only infer that the residual discrepancies from FourtEer’s formula 
shown in Table X. are not with any probability attributable to variations of con- 
ductivity 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 con- 
fidence in the discrepancies we have found, as true expressions of the deviations 
in the natural circumstances from Fourt£r’s conditions, a careful estimate of the 
probable or possible amount of error in the observed 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 
Forzes in the following terms :— 
The position chosen for placing the thermometer was below the surface “ in 
the Observatory enclosure on the Calton Hill, at a height of 350 feet above the 
sea. The rock is a porphyritic trap, with a somewhat earthy basis, dull and 
tough fracture. The exact position is a few yards east of the little transit house. 
There are also other buildings in the neighbourhood. The ground rises slightly to 
~ 
WOM. XXII. PART If. o2Q 
