304 Proceedings of Royal Society of Edinburgh. [sess. 
where Y is the amplitude at the surface ( x — 0) and p p q are con- 
stants, of which p and p should have the same value. The con- 
stant p is calculated at once by taking the ratio of any two of the 
amplitudes, and dividing the Napierian logarithm of this ratio by 
the difference of depth of the corresponding thermometers. The 
three values of p found in this way by combining the 1st and 2nd, 
the 2nd and 3rd, and the 3rd and 4th, are 0 , 00436, 0*00386, and 
0*00363, giving a mean of 0*00392. 
Then p may he calculated from the phases when the expression 
A cos 0 + B sin 0 is thrown into the form P cos ( 6 -f Q) ; for this 
quantity Q must be equal to -px + q. We have four equations to 
determine two quantities. Working them out by the method of 
least squares, we find 
p = 0*00371 2 = 0*9629. 
The difference between p and p' is not more than what might 
reasonably be expected. 
Finally, calculating the value of Y from each set, we get the 
four values 10*34, 10*35, 10*03, and 11*2, a very satisfactory 
result, giving a mean of 10*48. 
Hence we may write the most important term representing the 
annual wave of temperature passing downwards into the rock of 
the Calton Hill in the form 
v = 10-48 C-®""* cos (^t - 0-00371a; + 0-963). 
This gives a wave-length of about 16*93 metres, but before this 
depth is reached the amplitude of the variation has become too 
small to he appreciable. 
In the expression just given x is measured in centimetres. If, 
then, we integrate it with regard to dx from x equal to zero to x 
equal to infinity, and multiply the result by the thermal capacity 
of unit volume of the rock, we shall obtain an estimate of the 
quantity of heat which, at a given instant, is contained in the rock 
per square centimetre of surface. The value is 
cY f , /2 tt£ \ . Mt \ ) 
C 0 S It +.?)+p s 1 H-t vj 
where c is the thermal capacity per unit volume. 
