Mr. Ivory on the astronomical refractions. 461 
It will presently be shown that the other terms of the ex- 
pansion may be neglected ; because/ — x is very small ; and 
because the function multiplied by/ becomes inconsiderable 
after integration. Since we want only the definite integrals 
between the limits u = 0 and u = 00 , by applying the me- 
thod already used, we shall obtain, 
= -fa) Sin. flx [ f~ U 
— (/ x) . r in . d - c ~ u ( i •— c~ m ) 
A dii 
if f*du d.c~ u (c ~ u — i + w) 1 
* du J 
or, which is the same thing, 
<5'0 = «(i + a) Sin. 0 
*{.yv“ 
(/ > j 'J *du{2c~ zu — c~ M ) 
+jf 
a«(zc““— 2C 
0 
} 
In order to estimate the error produced by the terms of 
the expansion left out, we may compare the amount of the 
foregoing formula at the horizon, with the exact value of 
the horizontal refraction already computed. Now, when 
Sin. 0=i, Cos. 0 = o, A = V / 2 zw; and the expression will 
become, 
_ a (i + a) J" f*duc U / f ^ /* c 
— u X > J ✓ « 
+// 
du(zc' 
*du (2 c "" 2 M — c ~ M ) 
2 C- 2 U -.UC~ U ) 
V, 
} 
And if we now make u = /*, and then integrate between the 
limits t = 0 , t = co , we shall get 
( 1 + «) v/ w .. r 
