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 different 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 phaenomena which 

 we have been just now considering, can produce no change in the mean 



present these bases of calculation would be inadmissible. Experiments on po- 

 larization 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 tvhole 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 diflerent heights of the atmo- 

 sphere less than they had been found by the old method. 



AT. liiot's Bemarks 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. 



Vol. I.— Part III. 2 e 



