SUPERIOR REGIONS OF THE EARTH'S ATMOSPHERE. 395 
an excessive tenuity of matter the reflection is perceptible. Thus at 
that distance from the earth, the refringent power of the atmosphere 
ceases ; and, this being granted, we may calculate the angle that each 
luminous trajectory forms at this point of emergence, with its radius 
vector. We then see that this angle continues to decrease from the 
greatest elevations at which it had ceased to be perceptible by the senses. 
However the extreme value of it is but slightly ditferent from the last 
that can be observed; a fact which, when coupled with the free commu- 
nication of the layers of the air with one another, does not allow us to 
suppose that the intermediate values depart abruptly, or in any consi- 
derable degree, from the order of magnitude assigned by these two 
limits. If we then consider any luminous trajectory whatever reaching 
an observer placed at the level of the sea, under a certain zenith di- 
stance, which will also be the angle formed by the trajectory with its 
radius vector at the point, we shall be obliged to admit that while the 
former ascends to a very great height in the atmosphere, its element 
becomes more and more inclined to the latter. The subsequent vari- 
ations of this inclination, of whatever kind they may be, are always very 
inconsiderable, and lead to an ultimate value which, though lower, does 
not differ much from the values found at an inferior elevation. 
This condition, which is proper to the terrestrial atmosphere, being 
introduced into the general differential equations, conjointly with the 
values of the refringent power, the temperature, the pressure and the 
hygrometrical state of the inferior layer, I rigorously deduce from them 
for each zenith distance two values of the total refraction; the one ne- 
cessarily too high, and the other too low: so that the mean error is al- 
ways less than half their difference. Now when the latter becomes 
inappreciable by observation, the total refraction is found indepen- 
dently of every hypothesis respecting the uniformity of constitution 
and constancy of the refringent power of the superior layers, which are 
virtually placed beyond the reach of our examination; for then all 
possible diversities of condition, compatible with the phenomena which 
we have been just now considering, can produce no change in the mean 
“ote these bases of calculation would be inadmissible. Experiments on po- 
arization have shown in fact that multiple reflections contribute materially to 
the dissemination of solar light in the atmosphere ; and that, in each direction, 
rays reflected several times form a considerable part of the whcle bundle that 
_ reaches the eye. As to the rest, it is evident, that by introducing this new 
datum into the calculation, we should find the different heights of the atmo- 
sphere less than they had been found by the old method. 
M. Biot’s Remarks on this Note.—The less the thickness of the atmosphere, 
the more rapidly convergent are the developments of the refractions, and the 
more confined the definite limits which I have found for the refraction. Ac- 
cording to the result announced by M. Arago, those limits will become more 
narrow than they are given by the numbers which I adopted from the evalua- 
tion of Delambre, and consequently the zenith distances to which those limits 
may be applied can be extended still further. 
Vor. I.—Parrt III. QE 
