396 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 

 Ave take as our term of comparison the very perfect table of Mr. Ivory. 



We may calculate in a similar manner the refractions observable in 

 everj' 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, tliey 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 yojj^ 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 affoid 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 



