437 
and on Astronomical Refractions. 
nish so complete a test of the accuracy of any formula professing to 
give the relation between the pressure and the temperature in ela- 
stic fluids, as observations of the temperature of the vapour of water 
and other substances, which can be carried through a greater range 
of the thermometric scale, and above all through the low pressures 
where the character of the curve is more decided. 
M. Biot has dwelt with reason upon the importance of intro- 
ducing into the theory of Astronomical Refractions a greater con- 
formity with the conditions of the problem than has hitherto been 
attempted : and he has also noticed the imperfection in principle 
of the present mode of calculating heights by observations of the 
barometer, a method which must of course be abandoned (at least 
in any accurate exposition of this theory) whenever the discovery 
of the true connexion between the temperature and the pressure of 
the higher regions of the atmosphere renders it possible to adopt 
a more rigorous mode of eliminating the density from the dilfer- 
ential equation which connects d p and d The correct expres- 
sion which connects the difference of altitude with the pressures at 
the upper and lower stations ought to be the foundation of the 
theory of Refractions. Considering on the one hand the notions 
upon which my formula is ultimately founded, its identity with 
the results offered by the observations of steam and other vapours, 
and moreover the agreement afforded by the direct comparison 
with the observations of M. Gay Lussac, there can be no doubt 
that it represents the density of the atmosphere at different altitudes 
with greater fidelity than any hypothesis which has up to the 
present time been made the basis of the theory of Astronomical 
Refractions. 
I think that my table of mean refractions represents the observed 
quantities within the limits of their probable errors, and I have 
obtained this result without any arbitrary alterations of the con- 
stants. 
In the higher regions of the atmosphere the cold is intense*, 
depriving the air of its elasticity and converting it into a liquid 
or solid substance. My formula of course is only applicable 
as long as the air continues in the state of an elastic vapour ; 
and if at any altitude it ceases to maintain that condition, the 
density must be represented by a discontinuous function. But the 
density of this frozen air must be extremely small, and it probably 
has little effect upon the amount of Refraction. 
I am indebted to Mr. Russell for his kind assistance in the nu- 
merical calculations which accompany this treatise. 
29, Eaton Place, March 2, 1840. 
* See Poisson, Theorie de la Chaleur, p, 460. 
