Mr. Ivory on the astronomical refractions. 
42 5 
and in it to substitute s for x. Now, s = 
1 (1 4 04 ) 
— ' a —— if therefore we put i — 1 (I + - ^ , we shall get 
1 4- Sr') r a 5 0 
/(i + eo 
is = —, and 
dr : 
a dcj Sin. 0 
1 — 2 a a * \/ Cos . 2, 0 + 2 i.y — 
and, by expanding 
I — 2 ct cj’ 
dr=u Sin. 6 x 
d ai 
>/ Cos. 2 0 -f 2 is 2 0.01 
-f « 2 Sin. 0 X 7 — 
1 \4 Cos. 2 0 4 2 i s " 2 cooi 
+ &c. 
The second term of this expansion has to the first a less pro- 
portion than that of « to 1 , while u increases from o to i; 
and a greater proportion, while u increases from \ to 1 : 
and hence, on account of the smallness of «, we may com- 
bine both terms in one, viz. 
dr •=. a ( 1 -|- «) Sin. 6 v -/■- -- ^ (C) 
V 1 ' \/ Cos. 2 0 4 21S—2CC01 y ’ 
4. In order to appreciate justly the several formulae on 
which this theory depends, it is necessary to know the values 
of the quantities that must be found from observation. Of 
these, the coefficient a has been determined both astronomi- 
cally, and by direct experiments on the refractive power of 
the air. From the comparison of a great number of astro- 
nomical observations, De Lambre found K p = • 000588094, 
at the temperature of melting ice, and the mercury in the 
barometer standing at 29.921 English inches. In the same 
circumstances, M. M. Biot and ARAGO,by very accurate expe- 
riments on the refraction of air inclosed in a prism, found 
,000588768 for the value of the same quantity. Adopting 
. mdcccxxiii. 3 I 
