446 Mr. Ivory on the astronomical refractions. 
But in the formula of No. 7, when m = 4, a = 5 — x, 
= 1 —-(1 z) 3 ; and if we perform the operations indi- 
cated, and for the sake of brevity put 
_(i) r dz{i — z)i 
Q =4/— — 
Q (2 >= 
^(3> r dz(i—z ) 9 
Q = ioy — x — 
^(4) r dz{i — z )' 1 
Q = is/ 1 
we shall get, 
r=«(i +«)Sin. (x {Q (,) + x.-^.(_ 0 <')+Q (2) ) 
_j_ £_ _4_ . 5 6 (l) ~~ 
' 1.2' 5 — x 
i6Q^ + 11 Q^3) 
-f — • 
1 I . 2 . Z 
3 5 — * 
5 — * 
3(2) 
30 + 216 Q^ 2 ^— 396 210 Q (4) 
-f- &C. 
This is the general value of the refraction when m= 4- but. 
at the horizon, we get 
( 0 . 
Q 
Q 
o (3) = 
V z i (5 — A) 
2 
Vzi ( 5 - — a) 
v' 2 i (5 — a) 
Q 
,(4). 
x 4 fdt(i — ry = 
X lfdt(i - r ) 6 = 
X 10/rff (1 — f ) 9 
X 13/ dt ( 1 Z 2 )' 2 
2 64 
— ■ ■ 1 )( 
^2 i (5 . — a) 35 
2 1024 
v' 2 i (5 — a) 4 2 9 
1 3 1 °7 2 
V 2 i (5 — a) 46189 
2 „ 4*943°4 . 
V 2 i ( s — a) 1300075 
^2? (s— a) 
and, with these values, the series for the horizontal refraction 
will become 
r = — =^=Lx 5 1.82857 + x x 0.46717 
^2 ? ( 5 »A) 1 
+ x 2 x 0.18959 
+ X 3 x 0.08836 
+ &c. 
