394 M. BIOT ON THE CONSTITUTION OF THE 
density,—two conditions which already hypothetically limit the pro- 
blem. Besides, as the analytical integrals deduced from those equa- 
tions embraced the whole extent of the supposed atmosphere, they 
rendered it impossible to distinguish the share which the superior layers 
had in the total refraction obtained, and more particularly those shares 
which should necessarily be ascribed to them. 
The first of these difficulties has been solved in the memoir on Aséro- 
nomical Refractions presented by me some months ago to the Aca- 
demy. Tue differential equations of the motion of light in a spherical 
atmosphere, however constituted, are there established. But the ter- 
restrial atmosphere presents certain general phenomena which should 
be introduced into the equations, and which serve to limit them. Thus, 
the absolute smallness of the refractions at every distance from the 
zenith, excludes the possibility of the luminous trajectories which re- 
enter into themselves, and shows that those trajectories are all but very 
slightly curved. 
Again, at the elevations which are accessible to us, we find by expe- 
riment that the refringent power decreases in proportion as we ascend, 
and that the depression of the visible horizon increases. Hence, it is 
concluded that as we ascend from the surface of the earth, the angles 
formed by the elements of each trajectory, with their central radius 
vector, gradually diminish. This species of inflection is virtually proved, 
at the height even of Chimborazo, where Bourguer observed refrac- 
tions near the horizon; for, by a general theorem, given in my former 
memoir, the direction in which these refractions vary is found geome- 
trically connected with the mode of inflection in question. Analysis 
now shows that this geometrical phenomenon results from the feeble- 
ness of the refringent power as compared with the density ; and since, 
in the whole of that part of the atmosphere which is placed within our 
reach, the relation of these two elements is far from having attained 
the limit at which the phenomenon should cease, it is a necessary con- 
sequence of the law of the diffusion of the gases that it should exist at 
much greater elevations. In fine, even when we can no longer prove 
that it exists, we are able at least to assign limits beyond which it can- 
not extend: these limits are the result of the small altitude of the ter- 
restrial atmosphere. The observation of the twilights proves that the 
height of the last particles of air which afford us a perceptible reflection 
of light does not exceed, even if it equals, 7325 of the earth’s semi- 
diameter *, and the phenomena of the tails of comets show with what 
* Note by M. Arago.—All the determinations of the height of the atmo- 
sphere hitherto effected by observing the duration of twilight, rest on the sup- 
position that all the solar rays which mark the limit of the phenomenon have 
been but once reflected, and that, after being twice reflected on laminz of air, 
solar light is too feeble to produce more than an inappreciable glimmering. At 
