64 
9. The products of the alternate sides of the circumscribed 
hexagon are equal, viz : Af B y C h = A g B h Cf 
10. The properties in (8) and (9) hold good in the more 
general case ; that is, when the angles A C A 2 = B C B 3 ; 
C A C 3 = B A B, ; A B A 3 = C B C 2 ; and these angles may 
be taken either internally or externally with respect to the 
original triangle A B C. 
Many other relations might be deduced from the appro- 
priate diagrams; but the preceding appear to be the most 
important. 
A Paper was read by W. Fairbairn, LL.D., &c., “ On 
the Temperature of the Earth’s Crust, as exhibited by Ther- 
mometrical Observations obtained during the sinking of the 
Deep Mine at Dukinfield.” 
During the prosecution of researches on the conductivity 
and fusion of various substances, an opportunity occurred of 
ascertaining by direct experiments, under favourable circum- 
stances, the increase of temperature in the crust of the earth. 
This was obtained by means of thermometers placed in bore- 
holes, at various depths, during the sinking of one of the 
deepest mines in England, namely, the coal mine belonging 
to F. D. Astley, Esq., at Dukinfield, which has been sunk to a 
depth of 700 yards. 
The increase of temperature in descending, shown by these 
observations, is irregular ; nor is this to be wondered at, if we 
consider the difficulties of the enquiry and the sources of error 
in assuming the temperature in a single bore-hole, as the mean 
temperature of the stratum. At the same time it is not 
probable that the temperature in the mine-shaft influenced the 
results. The rate of increase has been shown in previous 
experiments to be directly as the depth, and this is confirmed 
by these experiments. The amount of increase is from 51° F. 
to 57f°, as the depth increases from 5§ to 231 yards, or 1° in 
99 feet ; but, in this case, the higher temperature is not very 
