420 
Mr. Ivory on the astronomical refractions. 
the vertical of the observer, then y — a Sin. 9: again, if ^ 
denote the density of the air at the surface of the earth, we 
get, by the formula before investigated, v = -y / 1 -f- K p, 
v = Vi+K p : wherefore 
y = - x/ = flSin. 0 x ; 
y v J \r i -f Kp ’ 
and, if we put a = l r> 
e 
y = a Sm.Qx^ /L+% ~. 
y V 1+ Kp'-Km 7 
and finally, 
I+Kp" 
a Sin. 0 
y ^ >/ 1 — 2a.a 
Let this value of y be substituted in the expression of dr al- 
ready obtained, then 
dr= * d “ x — - Sin - 8 
1 2 a w X. X \ 
Y/ (*+ — j 2 (i — 2 uu) — Sin. 2 0 
Whatever opinion we adopt concerning the height of the 
atmosphere, -L may be considered as a very small quantity. 
For, in every hypothesis, the density of the air is attenuated 
so fast in ascending, that it may be taken as evanescent at an 
altitude extremely small in proportion to the earth's radius. 
The quantity a is also a very small fraction ; and hence it 
will be sufficiently accurate if, in the foregoing expression, 
we put 
( 1 + ~) ( 1 — 2 uco) = l -j-2 -f 2aco; 
by which means we obtain, 
J a da Sin. 0 
I — 2 a.u W . x 
V CoS. 2 9-f- 2 2 a.u 
a 
In this formula u and r increase in ascending above the origin 
