90 Mr. Ivory on the Theory of the Astronomical Refractions. 
put the integral Qg, taken indefinitely, in the following form, 
which it is not difficult to verify by diferentiating : 
\—e‘^ 
- ~\f 
^ e ’ 
e ,2 d X c~^ 
~A ^ 16 
/ 
edxc~^ 
(2^ .2^ 175 ^ m 
V 16 
16 
48 
185 ,125 . 125 4 
)•/ 
) 
e dx c~^ 
/185 125 
— 6 I — A — , 
. -- Q 125 
" V 16 12 ^ 48 ^ 
c-^ A / 95 
/95 5 
(24^ 12 
0 
25 
24 
X . £' 
) 
This being the indefinite integral, the value of Qg in the 
formula for the refractions will be obtained by putting x m 
= 10 ,* which gives 
. c 
e e 
and this value, as well as that of s, being substituted, the 
quantity sought will be expressed as follows : 
^ pe,2dxc~^‘^ 91 pedxc—^ 
Q3=-V — A — ^tJ 
+ 
[^1 
[ 16 -' 
1 ^ J 
,2\4 
+ 
/125 1 125 1 905 1 
■? " 16 r +i 8 • 7+ 
A 
125/1 — A® 
48 V 
905 
\dxc~^ 
48 c® 16 ^ 
125 1 125 1 
— ;5' + 
48 
^25 
" 16 
127 
48” 
305 1 305 125 . 125 . 
48 ^ ^ 48 48 ^ 48 
48 e^ ' 48 
The series equivalent to the integrals must now be substi- 
tuted, in order to express the quantity sought in terms con- 
taining the powers ot e. 
In the first place we have these three terms, each of which 
is zero when the exact values of Aj, Ag, &c. are substituted, 
viz. 
125 , . , . 1 
-g-(A.-l+c-").- 
r 175 125 -L 
