434 
Mr. Ivory on the astronomical refractions. 
Sin. r 
Sin. <p Cos. <p — Sin. <p \/Cos. a <p — 2 a 
\/ 1 — 2a 
Now r = Sin. r -J- ^ Sin. 3 r -f &c. ; and, at the horizon where r 
is greatest, all the terms of the series after the first will not 
amount to of a second : thus r = Sin. r, and 
^ __ Sin. <p Cos. <p — Sin. <p s/ Cos . a <p — 2 a _ 
\/ 1 — 2 a 
by expanding the radical quantity in the numerator, 
Tan.® f , T a 1 IT “ 3 1 r a 4 , „ 1 
r “ X 1 * + 2 • + 2 • + 8 • + &C . } • 
When Cos. 0 = o, Tan. <p = and — - 1 2 ■ = — i_ . 
and hence, even in this extreme case, the term last set down 
of the foregoing series, and all the following terms, may he 
rejected ; therefore, because i -{-Tan.* <p, we have 
a. — a 2 
2 
i* 3 
= ^7^ Tan. * + *=J=Tan .• , + ^ Tan.’ » ; 
and farther, by rejecting the very small quantities a 3 Tan. <p, 
a? Tan. 3 p, « 4 Tan. 5 (p, &c. we obtain, with sufficient accuracy, 
r = ( a -f | « 2 ) Tan. ^ Tan. 3 <p -j- ^ Tan. s <p ; 
and finally, by substituting the value of Tan. <p, 
(q + | a 1 ) Sin. 9 
r 
| Cos . 2 9 zi + i z (• 2 
, j a 2 Sin. 3 9 
i cf 
+ i 
| Cos . 2 9 -f- 2 Z -f- 2* | 2 
a 3 Sin. 5 9 
| Cos . 1 9 -f 2 i -f i r | "2 
If we put Sin. 0 = i, Cos. 9 == o, we shall obtain the hori- 
zontal refraction in the hypothesis of Cassini, viz. 
=j*.{ 1 + T« + -TT + T-£} 
