440 
Mr. Ivory on the astronomical refractions. 
, f dz d - (i — z) m + n ~ 1 
+J A zn-r dz ; 
and because the function without the sign of integration is 
evanescent at both the limits z=o, z= 1, we shall get, with 
regard to the definite integrals, 
(l _ z )™ + rc 
a 2 n + i 
d. (i — Z ) m + n -'^ 
dz 
By operating in like manner with the quantities ■ 1 _ 
A 2 71 3 
d. (i — z) m + 1 and l dd. (i - z ) 4 ,n 
dz A 2 n 5 
shall obtain, 
dz 2 
we 
dz d-^-z) m + n - l V __ I 
r dz 
J . 2 n — — i ‘ 
£? Z 
rfrf. (l _*)* + » — 
2 n- 
dz 2 
= ' . ..±. r dz . 
zn — 3 ta J a 2« — 53 
=_ l_ . 4.. r rf f_ 
2K — 5 ta J zn — 
dd.(i—z) m + n - 1 l n 
dz 5 
</z 3 
A —'5 
And if we continue the like operations till we come to the 
quantity, -d- ‘ ■ — , which is no longer divisi- 
ble by %J/ ; and then combine all the results, we shall get, 
r dz( i-z) m + n ~ 1 4 , n 1 1 _ rdz 
J A 2re+I I.3.5...2B — I* i n n n J A 
I fdz d n .{\—z) m + n -'b n 
dz » 
By the application of this formula all the integrals in the 
value of r will be reduced to others in which the exponent of 
A is unit ; viz. 
