462 
Mr. Ivory on the astronomical refractions. 
^ ence 1 6 =20 55". 6. 
But we before found the horizontal refraction equal to 
2058". 5 ; and the difference is therefore no more than 2". 9, 
which is of no moment, since the refractions just at the horizon 
are alone affected by the error. 
In order to reduce the expression of £ (5 to the most simple 
form for calculation, we have, 
Ac-u-r- Cos. G = iC d ~ “ — Cos. 2 
the integrals commencing when u = 0 : and when they are 
extended to u = co , we get 
u ,uc 
~~A 
- u 
Cos.* 6 Pduc~ u , 
zi J * ” r 
Cos. 9 
2 i 
Now put 
N 
_ Cos SI flu- 
2 ij & 
Cos. 6 
m =2 zu —pim 
A A 
then by substitution we shall finally get 
2 2 
-U 
$ Q = oc ( 1 -f- a ) Sill 
+ x.M 
-/(2 M 
an 
in. « x \f% 
The expression of —j— may be put in this form, viz. 
a (1 + a) Sin 
480 
-■{*/* 
due' 
Cos.* 0 + 2 
>s.* 9 r d 
2 J {cos. 
fdl 
+ 2iy | ' 
ic- u (i-c- u ) 
Cos.* 0 + ziy\ : 
dH 
d r 
