132 Proceedings of the Royal Society 
distortion depending on the manner of diminution of the air’s 
density. 
In the actual case of the round earth, the phenomenon of a 
reflected image must necessarily lie between the above limit and 
horizontality. 
If a ray of light leave the point 
C, culminate at B, and reach the 
eye at A with a zenith distance, 
which we shall denote by A, the 
incidence at B must be 90°, and therefore we must have the 
equality 
OA.sin A. 1.000 294 = OB. 
or counting in English miles, and taking the earth’s radius at 3960, 
/ 
sin Ax 3961.167 = OB 
now the smallest possible value of the index of refraction /x^ is 
unit, hence the maximum value of OB is 
sin Ax 3961.167 = OB; 
wherefore, since the greatest possible value of sin A is also unit, 
it follows that the utmost limit of OB, under any circumstances 
whatever, is 3961.167 miles; that is to say, no culmination of light 
in the atmosphere can take place at a height of more than 1’167 
miles, or 6200 feet. But this light must reach the eye horizontally. 
If the angle A be less than 90°, the radius vector OB must be 
lessened in proportion to the sine; so that for A = 88° 38', OB 
becomes 3960, — that is to say, no inverted image can be seen at an 
elevation of 1° 22', unless the atmosphere have only an infinitely 
small thickness. 
We are thus brought quite to home. Our researches do not 
extend into the regions of the aurora and magnetic arc ; they are 
within the range to which our ordinary barometric measurements 
are known to apply with very considerable precision. 
This result comes from the known index of refraction of air, and 
from the earth’s dimensions, irrespective of all usual or unusual 
conditions of the air below the limit of culmination ; but it 
requires that at that limit the air should altogether cease. 
