471 
Mr. Ivory on the astronomical refractions. 
Having now found the values of the two integrals on 
which the expression of the refraction depends, we get im- 
mediately 
M= — L_ x j*e a x.M=— | o.02 7386.^ 3 
2 V I i S 2 \/ i l 
+ 0 . 234375 -e 5 
+ 0.250721.^ 7 
+ 0 . 194456 .^® 
+ o. 121089.^“ 
+ 0 .06s846.e 13 
+ 0. 029450. e '* 
4-0.012142 . e ' 7 
And again, we have 
N = 
but 
u 
4-0.051349.^’ 
4 - 0 - 054930 .e 7 
+ 0 .042602.^° 
+ 0 .026529^" 
4 - 0.013987^*’ 
-f- 0.00645 1.*? 11 
+ 0 .oo 2659 .<?' t 
Cos. 0 . 
2 i 
Cos. 1 0 m (1 — e*) 1 ___ (1 — e 2 ) 2 . Cos ,0 __ 1 1 — e 2 . 
2 j "4" ’ Ti e 2 ’ 2 i ~~ 2 y/J * 2 * e 
f*duc~ U * 
and hence, if we puty x ^ ; so that %{/ stands for 
the series in the value of the integral ; we shall get 
x 2 
Wherefore, by substituting the value of \|/, viz. 
4 = e 4- £ * 3 4--^ ** 4- A (3) . e 7 4- A (4) . e 9 4 - &c. 
we shall find 
N = ~rf x { — f <? — ie 3 4- 2 (4 — |- 4 - A (3) )-* 5 
4- 2 ( t 7 ¥ _2A (3) 4-A (4) ).^ 7 
4_ 2 (A (3) — 2A (4) 4-a ( s) ).^ 
4-2 (A (4) — 2A (5) 4-A (6) ).e M 
4_ 2 (A (5) — 2A (6J 4 -a (7) ).^' 3 
4-2(A (6) — 2 A (7) * ).e ' 5 
4-2(A (7) * * ).e 17 
