167 



and enable us to obtain the quantity of refraction, as changed by 

 the weight and temperature of the atmosphere, in which, near the 

 horizon, the French Tables appear entirely to fail. 



The first Tables in which tins has been attended to, as far as the 

 horizon, if I mistake nut, were those of Mr. Bessel.* 



In our ignorance of the law of variation of density, we can only 

 verify any hypothesis that we adopt, by a comparison of its results 

 with those obtained by direct observation. In this way, by help of 

 Dr. Bradley's observations, Mr. Bessel has obtained a modification 

 of the law of uniform temperature, that will give the refractions, to 

 within about three degrees of the horizon, with great exactness. 

 Dr. Young has, by adopting a law of variation of temperature ad- 

 vanced by Professor Leslie, obtained an equation for refraction, the 

 solution of which gives the refractions with considerable exactness 

 as far as the horizon. 



The following method is derived from the formula obtained in the 

 hypothesis of a density decreasing uniformly. 



It is shown in my paper, f above refered to, that, on this hypo- 

 thesis, 



Refr. =':L=J_tan [ 6 -(— i) refr 1 



Where d = zenith distance, / = height of uniform atmosphere, a 

 = radius of earth, and m : 1 : : sin incid. : sin refr. for air at the 



surface ; or making S =(^-^— ^^ i) refr. we may suppose the 



true refraction =-£=4 tan (0 — k S), the multiplier k as well as S 

 being different for each zenith distance. 



j-v,i- i\ 7 c r,' r, / ^ r\'n — I. ./ f i m — 1\ tan ^ 



sin I 



making 6 _ /l- S = 9', S = f 7^ r-. - ^T-^ tan / = (I Jli:l\ 



and the true refraction = ("'~ ' ^ tan F ^ — * ^"" ^ ( 1 '" ~ ' ") 1 



sill 1" •- sin l" ^ a , 2 -' J 



• Bessel's Fundamenta Astron, p. 26, &c. f Trans. R. I. Academy, Vol. 12, p. Qi. 



