474 
Mr. Ivory on the astronomical refractions. 
and hence, by substitution, 
£0=2 (1 2) Tan . 0 x 1 1 — o. 222614 • e z 
— o . 311358 . e 4 
• — o . 252551 . e 6 
— o . 142972 . e* 
— o . 057919 . e'° 
— o . 015603 . e .** 
Log. 2 ( 1 + 2) = 1 • 7671011. 
This transformation can be of use only to a certain distance 
from the zenith ; for at the horizon Tan. 0 is infinite, and 
the factor 1 — e* is equal to zero. The expression set down 
is sufficient for finding the refractions exact to of a se- 
cond as far as 85° from the zenith. 
And, if we take the logarithms of both sides of the last 
expression, we shall get 
Log. £0 = Log. Tan. 0 -f- 1 . 76710 
— o . 096680 . e 2 
— o . 145982 . e 4 
— o . 141413 . e 6 
— o . 114530 . e 8 
■ — o . 089474 . e'° 
- — o . 073278 . e tz 
which formula is very convenient near the zenith, and is suf- 
ficient for finding the logarithms of the refractions exact to 
five figures, as far as 84° from the zenith. It is to be ob- 
served, that while 0 increases from zero, e increases from a 
limit, from which it varies very little till 0 becomes a con- 
siderable arc. 
In order to have and it is only requisite to substi- 
