Gradation of Heat in the Atmosphere. 29 
However highly we may esteem the ingenuity and sagacity 
displayed in the experimental investigation of the formula, it 
seems hardly possible to ascertain, from that process alone, the 
degree of confidence that ought to be placed in its accuracy. The 
circumstances attending the experiments are such as to make it 
more wonderful that the author has been able to deduce from 
them a result at all conformable to nature, than one of a doubt- 
ful character only, and requiring to be confirmed by comparing 
it with actual observation. The decision to which such a com- 
parison has led seems to be this; that the formula is pretty ac- 
curate for small elevations, but that, in the case of greater, it 
determines the difference of temperature considerably above the 
truth. It agrees with observation at the earth’s surface ; but, as 
we ascend in the atmosphere, it makes the increments of altitude 
corresponding to a given difference of temperature decrease too 
swiftly. 
On account of the great simplicity of the formula, it would be 
very useful in many researches, if its claim to accuracy were esta-~ 
blished in a satisfactory, manner. It would, for instance, supply 
a desideratum in the problem of the atmospherical refractions, 
for which purpose indeed it has already been applied. Having 
entertained the idea of comparing it with the common method 
of calculating heights by the barometer, I find that by this means 
a criterion may be obtained that will enable us to form a correct 
opinion on the point in question. 
Using the same letters as before to denote the barometrical 
pressures at the bottom and top of a column of air, the height of 
which in fathoms is equal to 2; and neglecting the correction 
depending on temperature as unnecessary in the present inquiry, 
we get by the usual rule, 
10000 x log. = ag 
In this formula the logarithms are eval the common sort; and, 
as the ratio of the common to the hyperbolic logarithms is that 
of Re to 10000, we have 10000 x les: % = 4343 x hyp. 
log. = —; wherefore, 
Fi 
b 
Hyp. log. | re ee 
For the sake of abridging, ie C= a ; then, c being the 
base of the hyperbolic logarithms, we fuhaly et these equations, 
Viz. 
* Professor Playfair’s Outlines, § 341, p. 247, vol. i. 
Hyp. 
