Mr. Ivory on the astronomical refractions. 
44 3 
Hence, 
p fd z ( r — z) p 1 — , , „ 2 e . . p du . ( l — u) p *. (i e z u) 
a “S z i a 
P— 1 
consequently, 
p 'J' d z ^ 1 
2 e 
x ^pfdu(i—u)V 
— i 
\/ 2 2 a 
-f- e 2 • /> -p — l .f dn . u(i —u) v ~~ 1 
+ e 4 • p ■ p — i . ./ du .u*(i— u) p ~~ x 
+ &c. 
and, by integrating between the limits u — o,u~ l , 
p f 'dz{i-z)P - 1 __ c ; p ~ 1 c 3 I 
^ a v/77T‘ t i ^ 
p — 1 . p — 2 
5 +&c.}. 
A ^2ia ’ 1 1 P + i 1 p+ i -P a + 2 
This series will stop when p is a whole number ; and e being 
always less than l , it will converge fast unless when p is a. 
very great number. 
9. The horizontal refraction has not been determined by 
astronomers with much exactness. The quantity most gene- 
rally adopted is 33' 46". 3, which is that of the French tables, 
and is very little different from the determination of Brad- 
ley : it supposes the mean temperature of 50° of Fahrenheit 
and the barometrical pressure equal to 29.92 English inches. 
At the same temperature, and with the mean pressure 30 
inches, it is equal to 
203 i"- 5 . 
If we would compare with this the horizontal refraction in 
the hypothesis of Cassini, we have only to substitute in the 
formula found in No. 6, the values of x and i given in No. 4 : 
the result will come out equal to 
I2i8".6. 
The case, when the density decreases in the same propor- 
tion that the altitude increases, corresponds to m — 1 in the 
