( 588 ) 



„ = _^_ d-^- ^±n (VII1) 



or approximately 



ds = 



d l/tu 



l/4-l/IS-" + ' 



— f I 

 to 



Il is evident thai for small values of w the coefficient of e7|/coin 



(VIII) is smaller than that of da) in (I), and that also the refraction 

 in the lower strata will be found more accurately by means of the 

 formula (VIII) than by means of (I). For if we increase \/m in 

 formula (VIII) and to in formula (1) with equal quantities, beginning 

 with r /ero, we find, that, from m = to to = 0,2, the number of 

 values in the first case is twice as great as in the second case, hence 

 the integration by means of quadrature will give more accurate 

 results in the first case. 



Therefore I have used the formula (VIII) and computed the coef- 

 ficient of (IV a) for values of [^o, 0, 0,05. 0,10, 0,15 ... to 0,95. 



The density of the air which corresponds to [/a> = 0,95, occurs 

 at the height of about 18 kil. From the observations at my disposal 

 I could not deduce reliable values for the temperature at heights 

 above 16 kil.; yet it is probable that the gradients at those heights 

 are small and I have assumed the temperature at heights of 17 and 

 18 kil. to be equal to that at a height of 1(5 kilometers. 



In this way I have determined by means of mechanic quadrature, 

 and an approximate computation of the refraction between [/to = 0.925 

 and I to = 0.95, the differences As of the two values of the refraction 

 corresponding to Ivory's theory and corresponding to the table of 

 temperature 11 in the part of the atmosphere between the earth's 

 surface and a layer at a height of about 18 kil. where |/to is 0.95. 



1 have worked out this comptuation for the zenith distances 85°, 

 86°, 87°, 88°, 88°30', 89°, 89°20', 89°40' and 90°. 



An investigation, made for the purpose, showed me that in for- 



lo lo 



inula (VIII) the terms — {y — 3eto) in the numerator and — — (y — 2sto)* 



in the denominator may be neglected for all zenith distances except 



2 = 90°; therefore I have taken them into account in the computation 

 of the horizontal refraction only. 



