$96 M. BIOT ON THE CONSTITUTION OF THE 
value, but one so minute that its amount can neither be measured, nor 
the fact of its existence established by our instruments. 
By applying this reasoning to the meteorological circumstances which 
present themselves at the level of the sea, when the pressure is 0™, 76 
and the temperature 10° of the centesimal thermometer, I find that all 
the varieties of constitution that can be assigned to the atmosphere of 
the earth do not cause in our mean value of the total refraction a va- 
riation amounting to the following quantities, namely, at 45° zenith di- 
stance, 0", 001; at 74°,0' 277; at 80°, 2 243. These limits increase 
in proportion as we descend toward the horizon; but so long as the 
trajectory is not excessively low, the shortness of its passage through 
the atmosphere, together with the smallness of its curvature, causes 
them to deviate nearly in the same degree from the true refraction, 
which is then found to differ but little from their mean. The zenith 
distance being, for instance, 86°30', the mean error is only 1" 32, if 
we take as our term of comparison the very perfect table of Mr. Ivory. 
We may calculate in a similar manner the refractions observable in 
every other layer of the atmosphere, their meteorological elements 
being given, and shall find analogous limits of their values. It is ne- 
cessary to observe, however, that in proportion as the station of the 
observer is more elevated, these limits approach each other more 
nearly, for equal zenith distances ; and their deviation may in that case 
be disregarded, though under the same zenith distance, they are by no 
means to be neglected, when the observer is at the level of the sea. It 
is by these means that I intend to effect the solution of the problem 
which I have proposed to myself. For if we consider, for example, 
the trajectory which arrives horizontally at the level of the sea, and 
cause it to re-ascend into the layers of the air, according to a law of 
decrease sufficiently exact to bring it back, without any supposable 
error, to the height at which the density is reduced to the hundredth 
part of its primitive value (about >;%55 of the earth’s semidiameter), 
the angle which it then forms with its radius vector has become so 
small that the part of the refraction produced on the remainder of its 
course may be so exactly appreciated by means of our limits, that it may 
safely be included among the observations made at the earth’s surface, 
for the error cannot amount to 0°15" for the whole refraction. The 
superior layers, from which this portion is derived, might therefore be 
constituted in any imaginable manner as to their densities and tem- 
peratures, and in a certain degree even as to their physical nature, 
without our ever perceiving any appreciable effect of these differences 
in the total refractions observed; and thus, reciprocally, the observed 
refractions afford no idea of those elevated regions of the atmosphere. 
All that remains then is, that we endeavour to discover a law of 
decrease in the densities and the temperatures, such as may represent 
