Mr. Ivory on the astronomical refractions. 
4 33 
shell of uniform density, reaching to a certain altitude, and 
possessed of the same weight with the real atmosphere. The 
height of this homogeneous stratum of air will therefore be 
equal to the quantity /, before investigated. Suppose that the 
light of a star is refracted at the upper surface of this atmo- 
sphere in a straight line directed to the eye of the observer, 
and making an angle 9 with the vertical line ; then the per- 
pendicular let fall upon the refracted ray from the earth's 
centre will be equal to a Sin. 9 : and, in a right-angled triangle, 
of which a + 1 is the hypothenuse, a Sin. 9 one side, and cp the 
angle at the top of the atmosphere opposite to that side, we 
have, 
a Sin. 0 Sin. 0 Sin. 0 
o* u Olll. V 
Sill. <p — —r — 
T a -}- l 
i + — 
a 
Cos, o = J^P s - a 6 + 2 » + » a 
I 
Sin.0 
i + i 
Tan. cp == 
V Cos. 2 0 + 2 i -\-i x 
It is manifest that is the angle of refraction ; and if r be the 
refraction, or the angle between the incident and refracted 
light, <p 4- r will be the angle of incidence : and Sin. ( <p + r) 
will be to Sin. cp, as the velocity of the light in air to the ve- 
locity in vacuo, that is, as Vi + K p' to l , or as — to 1 : 
J VI — 2 a. 
wherefore, 
sin - (* + »■)= vi= 
Cos. (<p + r) 
\/ C0S. a lp — 2 a 
\/ I — 2 a. 
But, 
Sin. r= Sim (cp + r) Cos. <p — Cos. (cp -{- r) Sin. cp ; 
therefore, 
3 K 
MDCCCXXIII. 
