Mr. Ivory on the astronomical refractions. 441 
In the extreme case when m is infinitely great, we have 
and, 1 — u== L '~ s ’ 
r=«(i+a) Sin. 9 x f r d _ sc ■-= ' 
.7 v Cos. a 0 + 2is—2i\(l C s ) 
and if we expand this expression and apply the like reason- 
ing as before, we shall obtain 
A = V Cos. 2 9 -f- 2 i s 
r — a ( 1 + a) Sin. 9 x j 
+ K f d i- 
. * 2 rds 
+ m-J A 
d . c — 5 ( 1 — c~ s ) 
d s 
d d . c — 1 — c -5 ) 2 
d s z 
fds d 3 .c— s (i — c ~ s ) 3 
'1.2.3 </ A ds 3 
“h &c. 
an expression which has already been given by Kramp and 
Laplace, and is no other than the limit of the foregoing for- 
mula when m is infinitely great. 
The calculation of the refractions is now reduced to such 
integrals as J- z ~ Z ^ P — , p being any number; and the 
valuing of these must next engage our attention. 
8. In the first place, when 0 = go°, as in the case of the 
refractions at the horizon, then Cos. 2 9 = 0 , and A = V <2 i a z : 
now, put z = t *, and 
fdz(l- Z )P- 1 _ l_.fdt(l 
A 1/ 2 i a 
the integral being taken between the limits t — o and t = 1 . 
When p is a whole number, 
J ' ' 3. 5. 7. ... 2 p — 1 
which will apply conveniently in all cases unless when p is a 
great number. 
3 L 
MDCCCXXIII. 
