442 Mr. Ivory on the astronomical refractions. 
When p is great, assume 
1 
t'= c P ~ l : 
then 
/»— * a J_ i I - 
~~ p-l 2 ' { p — I) 2 1“ 6 ' Q 9 — ) 3 2 4 • (p_ l) 4 “ ' 
and by extracting the square root. 
, 1 r , _i_ .r 3 . J_ 
VyZTT X £ ” 4 >— I 96 ’ (j> — 1) 1 
l_ xl j 79 3:9 &c. 1 
128 ' { p — i) 3 • 92160 ' ( p — 1) 4 )' 
Hence, 
fdtli n v-i __ fdxc-*' f x -fl-Lii 
■' ' ) ~—J s/ v _ ! I 1 4 * p— x * 96 0 J 
&c. j : 
now, the limits of the integrals being t — o, t= 1, and x — o, 
x = go ; we get 
-XL xj,_i.-L 
2 • Vp— 1 t 8 p — I 
i°5 1 , 1659 
fdt ( ! — 1 % y 
4, ii . ! __ L__ 1 l6 ?9- l &r I 
' 128 ip — i) z 1024 ’ ip — i) 3 * 32768 (/> — i) 4 ■ j 
By employing proper reductions, any proposed case may be 
brought to another in which this series will converge swiftly. 
In the next place, when Cos . 2 9 is not evanescent, put 
z = u — e 2 ( a — u * ) ; 
then, 
( 1 —z) p ~~ l == ( 1 — - u) p ~~ 1 . (1 + e z u) p ~~ 1 
A = V Cos.* 9 -f- 2 i a ( 1—6* ) w + 2 z a e 8 : 
in order to determine e, assume, 
A =±= Cos. 9 -J- e n V2 i a ; 
then, 
V^ 2 ia 
ze 
Cos. 0 1 — e 
dz 2 e 
A s/ 2 i a 
x da. 
