Mr. Ivory on the astronomical refractions. 
i C due u .y _|_ 2 ix . p iuc ~ U (i— C 
J | Cos . 1 0 + J j Cos. 1 0 + 2 iy 
( 30 p ) o *•«(* +«) Sin, e . ■ r iuc~ U (i—c~ U ) 
-f Cos . 2 0 + 2 1 « 1 2 
460 
*f «’ 
Let us now assume 
r — C 1 + ~) -^ + C T ^o) -jp- (30 — />) • 
£ 0 being the mean refraction at the apparent zenith distance 9 ; 
then, by equating the like parts of the equivalent expressions, 
we get, 
$0 = ot(i + a) Sin. 9 x f^—y- ~~~~ ~ ---- > 
^ 1 ’ J V / Cos. 2 0 + 2 
<O0 
fit ( 1 Sin 
if«c “ 3 / 
480 
Cos. 2, + 2 iy jf- 
d$0 
X a ( I + a) fdu .C W (l 
= — “ x /7 
J \ Cos. 2 0 4 
•f 2 ix 
bc “(1 — c -u ) 
*V | Cos. 2 0 4 2iy| 
~c-“) 
+ 2 iy j 2 
each of which expressions must be separately considered. 
Now we have 
y = u-\-(f — X ) . ( 1 " c “) — j[c u — 1 -fw); 
and if we substitute this value of y in the expression of $ 9, 
and then expand the radical quantity, retaining only the 
terms of the first order, we shall get, 
= v^Cos.* 9 + 2 i u 
$ 9 ~ a, ( 1 + a ) Sin. 9 x 
C -^=^ 
