Mr. Ivory on the astronomical refractions. 4 65 
But the integral between the limits mentioned, is equal to the 
difference of the two values of the same integral taken, the 
one between the limits z = o, z = co, and the other between 
the limits z == 1, z = co. Now, by writing 1 z for z, the 
expression 
dz[z — z* ) c , 
will be changed into 
, ^re -rei C . , , 2 >.re -mz 
( l).C .1 dz (Z + Z ) C ; 
and it is obvious that the value of the former between the 
limits z= 1, z= 00, is equal to the value of the latter be- 
tween the limits z = 0, z = co. It follows therefore from 
what has been said, that we shall have 
A (re) m° + 1 f r j / ^\ n ~ mz / \ n -m Cj / ■ .\ n -mz 1 
= ,- . 3 :.. n*\J dZ ( Z — z ) C — (— ' 1 ) - c .Jdz(z + z‘)c j, 
each of the integrals being extended from z — 0 to z — co. 
Again, p being any whole number, we have, between the 
limits z 02: 0, z= co, 
f 
dz.z p .c - mz = L±JL^L 
mP + 1 
Wherefore if we expand the binomial quantities in the value 
of A^, and integrate the terms separately, we shall obtain 
A (n) = i — 
n 
n + 1 
-j- n 
n — 1 » + 1 • #+2 
— (— 1 )“. c -’ l ‘.{ lJ f n - n -^r+ n 
■ n • 
n- 
1—1 n — 2 ra + i.ra + 2.w + 3 
2 3 m 3 
1 n + 1 . re -f- 2 
+ &C. 
4 -&c.). 
By this means we get the first part of the integral sought in 
a series that has all its terms positive, and that will always 
converge because e never exceeds unit. 
Let us next consider the second, or supplemental part, viz. 
n duc~ u 
J \/ Cos. 4 6 + zim -f ziu 
- m 
C X 
3 O 
MDCCCXXIII. 
