160 Dr. Young on a finite and exact expression for the refraction 
radius of the earth, and here dy = ^\/zdz — zdz, we have 
dx = — £ — I- — , and fdx= — A , for 
the height above the earth’s surface, which, when z = o, 
becomes \ x 27300 = 95550 feet. For the refraction, we 
have the equation d r— [1 - z ]) ( Astr - ColL XV ) 
= f d - — - r, which is the value originally as- 
signed to this fluxion by Dr. Brook Taylor ; v being the sine 
of the apparent altitude ; and here 
dr 
— p dz 
9,2 z \ ? 
-V/* + ~ +**-2^+2 pt l 
'{i- 
or, if z = \|/, and d z = 2 >}/ d 4/, 
dr ij/ d i]/ 
2 p 
which is equivalent to the 
,rdi 
of the Article 
\/ (a -J- b x -f- c x x) 
Fluents in the Encyclopaedia Britannica, No. 259 ; the fluent 
being 
\_V{a + b x + c x 2 ) — ~^- c h 1 (2 c x + h + 2 Vc V(a + h x + c <!)]» 
and its whole value, from z = 1 to z = o, being — ~ 
1 
= V (a! v*) — v fl — °- r h 1 - - 2 C + ~j ?y ° s , putting a 
= 2 p, since a + 6 -j- c = z> 2 . 
. For the numerical values of the coefficients, taking, at the 
temperature of 5o°,p = . 0002835, and — == . 001294 = 
772.8? 
a = — — 2/> + ^ 9 =. 008491 + t ) 2 J = — 2 . — — . 011646, 
m x 3 m 
and c = — 4 - 2 p = . 003155 ; hence — = . 17972, Vd 
m 1 r 
= . 0921466, = . 05617, 
= , 10367, 2 r -{- 6 = 
. 005336, and 
2 \/ C 
17972 (✓(■ 008491 + p») - v— . 10367 h 1 . 0 ,, 64 6Cnf 3 4v>l'o il5H^) 
