4<66 
Mr. Ivory on the astronomical refractions. 
Now Cos. 2 0 = — ~ xzim: and, if we substitute this va- 
lue, the integral will become 
and, by expanding the denominator and integrating between 
the limits u = o, u = a>, we shall get this value of the quantity 
sought, viz. 
This series will converge in its first terms : and the results 
being alternately too great and too small, we can thus esti- 
mate the degree of approximation. 
By uniting the two parts, we get this expression for the 
whole integral between the limits u = o, u = m being an 
arbitrary quantity, viz. 
