480 
Mr. Ivory on the astronomical refractions. 
instances, but which cannot fail to leave a final error in a 
numerous set of observations. 
We are in possession of another table of refractions, com- 
puted with great care, published in 1818, by M. Bessel, in 
his Astronomice Fundamenta. This table must be considered 
as having the authority of actual observation as far as 86° 
from the zenith ; since, to that extent, it represents with great 
exactness the observations of Dr. Bradley, which served as 
the basis of its construction. At lower altitudes, the refrac- 
tions in it are confessedly too great. To compare the new 
table with that of M. Bessel is, therefore, the same as to 
make a direct appeal to experience. 
The astronomical refractions were first discussed with a 
due attention to all the circumstances of the problem in the 
Treatise published by Kramp.* This author gives the name 
of specific elasticity to the quotient of the relative pressure 
divided by the relative density of the air ; it is therefore equal 
to l±_£j in the formulae of the present paper. He represents 
it by c~ eS , s being a small fraction ; which function, therefore, 
contains the law for the gradation of heat according to 
Kramp. Now, if we substitute c~ es for * * ^ ~ in the foregoing 
formulae, and then equate the two values of P, we shall get 
This is the rigorous expression of the density in the hypo- 
and hence 
£ S 
(C — l) + sS 
1 — u— C . 
* Analyse des Refractions Astronomiques et Terrestres, 1798. 
