of Edinburgh, Session 1882-83. 
131 
holds good independently of the variations between the points A 
and G, and of the thickness or thinness of the supposed layers. 
Hence if these three factors be known for any one point in the 
path, and if two of them be given for another point, the third 
factor corresponding to that point may be at once computed. 
In this part of the theory of atmospheric refraction there is no 
difficulty ; the astronomer’s trouble is in discovering the geocentric 
angle AOG. 
If the earth had been flat, the computations would have been 
still simpler ; the product of the sine of the zenith distance by the 
index of refraction would have been constant, independently of the 
height, and the astronomer would have deduced the true from the 
apparent place of a star by a simple proportion. 
M. Biot has given the index of refraction for air at 1.000 294. 
It varies with the pressure and temperature, and the formula 
h 
''"^■‘‘100 000 
in which h stands for the height of the barometer in English 
inches, is sufficiently near for our purpose, is, perhaps, within the 
limits of error of the observations. 
A ray of light rising obliquely from a point on the surface of a 
flat earth until it arrived at the upper limits of the air would have 
had the sine of its zenith distance augmented in the ratio of 
1.000 000 to 1.000 294 ; and if the apparent zenith distance at the 
outset had been 88° 38', that is, if its angle of elevation had been 
1° 22', whose secant is 1.000 294, the zenith distance at the upper 
surface of the air would have been 90°, Ho sun, moon, or star 
could have been seen at a lower altitude than V 22', Ail light 
reaching the eye from a lower elevation must have come from some 
terrestrial object after having culminated as at B. The images of 
terrestrial objects would have been seen constantly in the air, but 
distorted by the retroflection, — the amount and charactei: of the 
