447 
Mr. Ivory on the astronomical refractions. 
and by completing the calculation, we shall get, 
r = 2041". 3 - 
This result is very near 2031" 5, the horizontal refraction 
usually adopted ; the difference 9 ". 8 being much less than 
the uncertainty in the determination of this quantity. But it 
will be more satisfactory to compare the refractions in this 
hypothesis at all altitudes with those admitted by astronomers. 
In order to find a formula for this purpose we have only 
to substitute for Q^, Q^ 2 \ Q^\ the series investigated in 
No. 8 ; and we may leave out the term multiplied by A 3 , since 
the amount of it is less than 1" even at the horizon. Thus 
vve get, 
v' 2 i (5 — x ) 
Cos. 0 
+ 
+ 
2 a (1 + Sin. 0 
. . x \e — e 5 - 4 ~ — 
v'z* ( 5 _x) 1 ' 5 ”5 ' 35 
±ZL (JV + 11,5 r ISL -L.fi . il 
— * y 20 1 60 * 210 I 22 * 122 
5 
2 . x 
( 5 —*) 
Li£_ (-L e 5 _i_ 121 e 7 1 22 . p 9 t 12 ^ \ 
-»)*M 3 + w e 2 
And, by substituting the numerical values, we shall find, 
Tan. (p = 19.0462371 -f Sec. 9 — 20. 
</ Log. of Coeff. 
r = 1048.95 x Tan. \ (p Sin. 9 . . . 3.0207 558 
-f 658.21 x Tan . 3 \ (p Sin. 9 . . . 2.8183661 
+ 252.92 x Tan. s \ <p Sin. 9 . . . 2.4029800 
+ 59-64 x Tan . 7 \ <p Sin. 9 ... 1 .7755092 
+ 11.61 x Tan. 9 i <p Sin. 9 .. . 1.0648048 
4- 2.95 x Tan. M i <p Sin. 9 . . . 0.4706968 
But it is to be observed, that the logarithm of - 22 - has been 
0 29.921 
subtracted from the logarithm of every coefficient, in order 
to bring the formula to the same barometrical pressure with 
