Mr. Ivory on the astronomical refractions. 
439 
■z 
)■}: 
s = m + i • z, 
i—co=(i — z), m 
a = 1 — ( 1 — z), m 
2 is — 2 aw = 2 Z (m-f- 1 ) % — 2 1 — (l 
and, again, 
X j a 
a = m + i — x 
4 ,= ! — ( 1 — z) m -' 
2 i S 2 a 0) = 2 i 3. Z 2 i X ( 1 2 ) 
d co = m d z ( l — %) ™ \ 
The expression of the refraction (equation C) will therefore 
become, 
r=*( i + «) Sin. t X f.. : 
V 1 y V' COS . 4 0 2 2 <1 Z — 22^(l — 2)4 
and by expanding the radical quantity, 
A= t/ Cos. 2 9 -}- 2 z a 2 : 
r = « (i + «) Sin. 9 x | J™dz{i — z) m — 1 
A 3 
4 _ - 1—1 . z ‘ 2 x *. fmdz{ i—z) m + 
^ *•* J ^ 
-f &c. 
And, in this expression, it is not necessary to integrate ge- 
nerally, but merely to find the definite integrals between the 
limits 2 = 0 and 2 = 1 . 
Now by taking the fluxions of the quantity l — , 
we have 
(,_ z) « + «— i,j,» 
2 n — i 
■ 
dz{ \— z) m + n ~ 1 
A 2 w + 1 
