Mr. Ivory on the astronomical refractions. 
463 
and it will be sufficiently accurate to write u for y in the de- 
nominators of the two last terms : then 
I n . a ( I + «) Sin. 0 / Pduc~ u { 1-C~ U ) 
— 5 — • < l x 2 X / — - 
480 2 480 \ U A 
Cos. 2 0 Pduc~ u \ 
' ~ J ( : 
<m_ 
dr 
and, by the same procedure as before, 
d t 480 2 
But we have 
1 S' 0 . a ( i 4 “) Sin. 6 
T T 
480 
2XM 
Cos. 2 0 (~*duc 
-J 
Cos.0 
'due u 
A 7 " 
the integrals commencing when u = 0 ; and, when u = co, 
we get, 
Cos. 1 © pduc~ u _ 
J A3 -■ 
Wherefore, 
£?S0 I S0 , a(l + a) Sin. 0 
Cos .* 9 Pduc ~ u Cos.0 
+ 
dr 480 2 * 480 
and, by substituting the value of S 9, 
|2 X M + N j: 
dH 
a ( I + a) Sin 
480 
— x ’+I»m+n 
A UM - i 
Lastly by writing u for y in the expression of —yj, we readily 
obtain 
dS0 , a (i 4- a.) . X Sin. 0 A/r 
^7 = H xM - 
3° 
Thus the quantities $9, ultimately involve only 
two different integrals, vi U and J * dv £ * W ; the va- 
lues of which we must next endeavour to investigate. 
