Mr. Ivory on the astronomical refractions. 455 
* == l x { 1 + /3 • • Lo g- T’ 
which is no other than the usual formula for barometrical 
measurements, x being the height corresponding to the re- 
lative pressure P. 
Again, when m is infinite, we have already found, 
u 
1 u ■ — C , 
p=( 1 -/) c -“+/ c - 2 “= c -“{ i _/(i_ c -“)}, 
* = ( 1 — f ) « + */( 1 — 
t + |3 t P r 1 — 
TT ?7 = — = 1 -/( 1 - c )• 
Hence, 
L°g- t = k + log- 
and, neglecting the cube and higher powers of 1 — r M , 
Log.y = w +/(i “ C ~ W ) +T* (1— 
But, we easily get, 
1 + j 3 
T -f t' 
I + 
wherefore, by multiplying, 
I + ^ T + 4 
L 0 ®- T * ■ + = “ +/0 - c ~“) — 4 « 0 -‘~ u )- 
Now, because 
/«=»/(!_ e —) +4(1-C — )*+&C. 
we get, by substituting and neglecting the same powers of 
( 1 — c” w ) as before, 
T + 4 
' =«+/(! -C-“)_Z 
Log-T 
I + /3 
Farther, 
1 + j3t' 
s = u—fu + 2/(1 — ; 
