Mr. Ivory on the astronomical refractions. 
4 64* 
f* duc~ u 
13. The whole integral J , extending from 
u = 0 to u = co , is composed of these two parts, viz. 
duc~ w 
fx 
\A 
v/Cos. 1 9 -J- z i u 1 t/ V /> Cos. 2 9 + 2 im + 2 iu 
the first part being contained between the limits u = 0, and 
u = m ; and the second part, which arises from substituting 
m -j- u for # in the first part, being extended from u = 0 to 
u = 00 . 
To begin with the first part : put 
u — m (1 — e 2 ) % -j- m e 2 z 2 ; 
and the limits of u being 0 and m, the limits of z will be 0 
and 1. In order to determine e, assume 
A — V Cos. 2 0 -J- 2 im (1 — e 2 )z- f- 2 ime 2 z 1 — Cos. $-\-ez\/ v.i m : 
then 
s/ zim 
ze 
Cos. 0 
du z e 7 
— — 7 x md z, 
A V zim 
AiA “= Pmdz 
A V2 z m 
/ j — u 
~ between the limits u = 0 , 
u — m, be expressed in a series of this form, viz. 
'“"v/TTS" x | • « + A (1) e’ . . . . -f- A^c z " + 1 . . . &c.| 
then if we develope the foregoing exponential value in a series 
of the powers of e , and equate the like terms of the equiva 
lent expressions, we shall get, 
A W ~ ~k ■fdz(z-zTc- m *, 
the integral being taken between the limits z = 0, z = 1 . 
