220 
SECOND REPORT -1832. 
Several formulae have been proposed for a general expression 
of the law of gradation. Lagrange, and many after him, con¬ 
sidered the decrement of heat to be simply proportional to the 
height, and the best observations may be tolerably representedby 
t = n H. 
Where t , the decrement of temperature, is expressed in degrees 
of Fahrenheit, and H in English feet, the coefficient n will be 
a fraction nearly equal to 27 o> or between that and - 3 ^ 3 -, for 
small heights where the decrement proceeds most rapidly : the 
observations of Humboldt and those at the Grand St. Bernard 
w T ould, we have just seen, make n about jj-q. Euler considered 
the progression an harmonical one. Professor Leslie, from ex¬ 
periments upon the heat absorbed by air in rarefaction, pro¬ 
posed theoretically the formula 25 
for the diminu¬ 
tion of temperature on the centigrade scale ; where Q represents 
the density of the air at the upper station. This formula was 
first proposed without demonstration* * * § ; afterwards the nature 
of the experiments upon which it rested was explained f : from 
its principle, this formula only takes cognizance of the influence 
of the change of specific heat in the atmosphere, without any 
reference to the effect of the mass of the earth. Professor Les¬ 
lie’s formula has given rise to several discussions, to which it 
will only be necessary to refer in a note J. 
Mr. Atkinson, in discussing the subject of Astronomical Re¬ 
fractions §, has examined with great care all the actual obser¬ 
vations to which he had access, and from them he has deduced 
the following formula : 
h = [251*3 + § 0 - 1)] n, 
where h is the height of the station in English feet, n the de¬ 
pression of Fahrenheit’s thermometer. 
An ingenious attempt was made by M. Mathieu|| to deduce 
the law of decrease of temperature in the polar regions, by 
analysing two observations by M. Swanberg upon the amount 
of refraction, by means of the formula of the Mecanique Ce¬ 
leste . The result at which he arrived was 243 metres of eleva- 
* Leslie’s Elements of Geometry. 
t Encyclopedia Britannica, Supplement, Article Climate. See also Thomson 
On Heat, p. 122. 
I By Mr. Ivory, Philosophical Magazine, 1821; by an anonymous writer, 
Edinburgh Journal of Science, vol. v.; and a paper by myself relative to the 
last-mentioned one, Id. N.S. vol. v. 
§ Memoirs of the Astronomical Society, vol. ii. 
j) Humboldt, Observations Astronomiques, i. 155. 
