166 



This inquiry lias led to the extremely complex but elegant mathe- 

 matical researches of Kramp, Laplace, and Bessel. Tlieir investiga- 

 tions are nearly related to each other. Dr. Young* has also recently, 

 by an entirely different method, and with great analytical skill, ob- 

 tained an equation expressing the relation between the refractive 

 force of air and the refraction at any zenith distance. 



It is an objection to these inquiries, however curious in them- 

 selves, that the results are entirely useless for the nicer purposes of 

 astronomy, unless we shall also be able to reduce to calculation what 

 now appear as irregularities. 



To this it may be answered, that, when the difficulties as to the 

 regular refractions are overcome, the way will be cleared for an 

 attempt, apparently indeed more hopeless, on the others. 



It is admitted, that the true refraction is always less than that 

 computed on the hypothesis of an uniform temperature, and greater 

 than that obtained by supposing the density to decrease uniformly ; 

 that, as far as 80° from the zenith, these limits approach each other 

 very closely-f- ; and that, till very near the horizon, they do not widely 

 recede from each other. 



The former hypothesis has given occasion to elaborate investi- 

 gations by M. M. Kramp and Laplace, and a modification of it to 

 still more elaborate ones by M. Bessel. 



It is the object of tiiis paper to deduce, by help of a modification 

 of the result of the hypothesis of a density decreasing imitbrmly, by 

 an extremely simple investigation, the refraction, at any low altitude, 

 corresponding to any heights of the barometer and thermometer. 

 The Tables thence resulting for zenith distances, between 80° and 

 the horizon, will, I conceive, be foimd as convenient as can be 

 desired. They scarcely yield in simplicity to the French Tables, 



• Phil. Tranr. 1819, Part I. f Tians. R. I. Academy, vol. 12. p. 89. 



