Mr. Ivory on the astronomical refractions. 44 9 
elementary quantities in the algebraic formula is neglected. 
Had the computation been rigorously made, as Bessel has 
since done in the table he published in 1818, the mean re- 
fractions of the French table would have been less, instead 
of being greater, than the results of the foregoing formula. 
But it was the opinion of the eminent astronomers under 
whose direction the table was published, that the refractions 
near the horizon are too uncertain to require attention to 
minute accuracy. 
It appears therefore, as far as we can form an exact judg- 
ment, that the formula approaches very near the true mean 
refractions. It will afterwards be shown that the hypothesis 
from which it has been deduced, likewise represents, with 
considerable accuracy, the pressures and densities actually 
observed in the atmosphere at different heights. But in one 
respect there is a deviation from nature. According to the 
supposition m = 4, the total height of the atmosphere is equal 
to 5 x /, or about 25 miles, which, in all probability, is hardly 
equal to half the real height. It therefore becomes neces- 
sary to inquire what influence this circumstance will have 
on the quantity of the refractions. 
10 Continuing to represent the density, or 1 — «, by 
( 1 — z)", we may assume 
/being an arbitrary quantity. Then, from the formulas (A), 
we get 5 =) and ~ ; and hence 
s = (l—f) .z+zf. {1 
and it is to be observed, that this last quantity is always 
MDCCCXXIII. 3 M 
