472 Mr . Ivory on the astronomical refractions . 
And, in numbers, 
N~-^= x {- ie — j - e 3 +0.158714. e 5 
+ 0.216022 . e 7 
0.14401 2. e 9 
+ 0.068960. e" 
+ 0.026138 . e 13 
+ o. 007402. e li 
+ 0.003752 .£ 17 
To find $ 9 , it only remains to substitute the numerical 
fd ~ U 
values of J , x M, and N, in the expression investigated 
in No. 12 ; then, 
$ Q = x Sin. 0 x^x { 1 + 0.777386.^ 
+ (0.488849— 0.09 1286./). e * 
+ ( 0.259287 — o. 183242./). e 6 
+ (0.1 21827 — o. 205287/). e * 
+ (0.052628 — 0.160168./). £ t0 
+ (0.021440 — 0.097828/). e ' 2 
+ (0.008327 — o. 050560/). e' A 
+ (0.003081 — o. 020321/). e ' 6 
The two first terms of this expression do not contain/; 
and they give that part of the refractions near the zenith, 
which has no dependence upon the constitution of the atmo- 
sphere. As there is some uncertainty in the value of /, it 
may be determined either so as to make the horizontal re- 
fraction coincide with the quantity adopted by astronomers ; 
or so as to make the formula represent some very exact ob- 
servations made at low altitudes, from 2 0 to 7 0 above the 
horizon. With regard to altitudes less than 2 0 , it is not clear 
that the astronomical refractions do not participate of the 
extreme irregularity that attends the terrestrial refractions, 
