469 
Mr. Ivory on the astronomical refractions. 
+ 0.079225./ 
+ 0 .026099.^" 
+ o.oo7453-£‘* 
+ 0. 001876. e 15 
+ o. 000422. / 7 
Although we are sure that this value is a near approximation 
ther it be sufficiently exact for the purpose intended. Now, 
the part of $ 9 depending on this integral, is 
and, this being valued by means of the foregoing series in 
the case of Sin. 9 = 1, and e=i, in which circumstances the 
error of the approximation is greatest, the result will be 
But, when Sin. 9 = 1, A= ; and the quantity we are 
considering will become 
and if we put u = t % , and integrate between the limits t~o, 
t= co, the exact value will be equal to 
It appears, therefore, that the error of the approximation 
when it is greatest, or when 1=1, does not amount to half a 
second. But as the error is expressed by a series of terms 
multiplied by e 1 , e°, &c. it diminishes very rapidly as e de- 
creases, and becomes altogether insensible when 9 is less 
than 90°. 
to the truth, yet it may not be superfluous to examine whe- 
= 2037". 8 . 
