473 
Mr. Ivory on the astronomical refractions. 
which would render such observations unfit to be employed in 
this research. But in the present state of our knowledge it may 
be doubted, whether a more satisfactory determination of f 
can be obtained than what we have hitherto assumed, namely, 
f—j;. With this value, we get 
$6 = x Sin. 0 xex|i+o. 777386 . e 2 
-}- t> . 466028 . e 4 
✓ 
+ o . 213477 • e* 
+ o . 070505 . e 8 
-f- o . 012586 . e'° 
— o . 003017 . e lz 
— o . 004313 . e 14 
— o . 001999 • e ' 6 
X GC ( l -l- C&) 
LOg. V = 2 . 9101040- 
2 V i 
Tan. <p = 19 . 1580271 -j- Sec. 0 — 20 ; e = Tan. \ q>. 
If we make 0 = 90°, and e = 1 , we get, 
$ 9 = x 2 . 530653 = 2057". 5. 
2 V i 
This is the horizontal refraction by the formula : and as the 
exact value of the same quantity was before found equal to 
2058". 5, the error arising from the method of approximation 
amounts only to 1" at the horizon. But all the quantities 
neglected being of the orders e 7 , e 9 , &c. the error will be al- 
together insensible unless when e is extremely near 1, that 
is, at very low altitudes. 
The foregoing expression may be put in another form, 
which, in some cases, is more convenient for calculation. 
Since 
2 e 4 V i 
1 — e* Cos. 0 
; we get 
1 — e a 
Cos. 0 
3 P 
MDCCCXXIII. 
