44-4 Mr. Ivory on the astronomical refractions. 
formula of No. 7 ; and being = 0, we get, 
r = a ( 1 + * ) Sin. 6 ; 
and, at the horizon, 
r=«(i +«) . 
V z id. z S' 3 i a 
Now, a being equal to w + 1 — x, we have in this case 
a =ss 2— x; wherefore, 
r — 2a C 1 + a ]_ __ g (* + g ) — . 
Vzj(2 — *) \/ . X 
* 1 ”T 
In both these hypotheses, although the refractions near the 
zenith agree with nature, yet, at the horizon, they fall 
greatly short of observation. 
At the other extreme, when m is infinitely great, the term 
which is multiplied by x” in the expression of the refraction 
given in No. 7, is thus expressed, viz. 
** p d s d n . c 1 — c 5 ) 
but, at the horizon, A= x/^TT 5 therefore, 
d s d n . c~ s { 1 — c~~ 5 )” 
x n 1 d s 
- X —= X / - 7 = • 
1 . 2 . 3 .. .71 V 2 l J V 5 
d s" 
and, by expanding and performing the operations indicated, 
the same term will become 
1 fy d S f —5 — 25 tl 1 3 5 ^ -) 
x Vfi {±^ +n.2 n . c ±n. — -3 n c + &c. } 
the upper or lower sign taking place according as n is even 
or odd. If now we put s = f, and then integrate between 
the limits t = 0, t = 00 , we shall get, 
^ 7T f , 2 ” 
Tri * |± I+B - ✓r 
+ n. 
n — i 3 n 
= +&C. } 
I.2.3...W \/ 2 i l ~ ' V2 — * V 3 
Hence if, in the case of the horizontal refraction, we assume 
