on Atmospherical Refraction, &c. 197 
tudes, as found from observation, it will appear that the re- 
fractions vary, as the tangents of zenith distance, minus some 
multiple ofr (the refraction), which cally. Now the value 
of y will be found by the formula drawn from the investigation 
given at large in Mr. Vince’s Astronomy, by comparing the 
observed refraction, at different zenith distances, putting r = 
refraction, a = zenith distance of the highest star; r' and a' of 
the lowest star : then 
r x cot. a — r x cot. a! 
y — » ® 
Applying this formula to the mean refraction of Polaris below 
the pole =49", 105, compared with the corrected or observed 
refraction of each of the last three stars ; the mean value of 
y = 3,3625. Dr. Bradley found by comparing the refraction 
at 6o° with that at 90°, which he supposed 33' o", the value of 
^=2,996 ; he therefore assumed 3. M. Cassini found the same 
to be 3,226; M. Bouguer 3,323; and Mr. Simpson 2,75: this 
difference arises from their having supposed the horizontal re- 
fraction greater than it appears to be from observation. In the 
dissertation of the last author, he proves, that above y° alti- 
tude, it matters not whether you assume the refraction, ac- 
cording as it would be found from the increased density of 
the air, at the low altitudes, and which would give the refrac- 
tion at 90° about 52', or by an uniform density, which agrees 
better with observation ; since, in the former case, it would 
only affect the refraction about two seconds. 
Having thus assumed a greater value ofy, than 3, for the 
coefficient of r, the mean refraction at 45 0 will vary inversely, 
as tang, z — %r : 58", 10734 : : tang, z -~yr : x . therefore. 
x 
tang. 45°- 3 r x 58.10734 __ „„ 
tang. 45 0 — vr 5 > 9 ’ 
■yr 
D d 
MDCCCX. 
