506 Mr. Ivory o?i the Theory of the Astronomical Refractions, 



of the phsenomena of the atmosphere, it seems impossible to 

 determine f by experiments. The probability is, that it is 



2 

 much less than/" or - ; and as tiie integral Qg is inconsi- 

 derable except within a degree or two above the horizon, 

 and even at such low altitudes is not great; it follows that the 

 part of the refraction depending onf Qg will only be sensible, 

 if at all, when a star is distant 88° or more from the zenith. 

 At present the probability is,' that there is no other way of 

 ascertaining the value of/"' but by good observed refractions 

 at great distances from the zenith ; which observations are 

 neither numerous nor easily collected. From the uncertainty 

 of the tevmf X Qg , it cannot be estimated in constructing 

 a table of mean refractions, which must therefore be deduced 

 entirely from the other three terms, as in the paper of 1823. 

 In this manner has the table in this paper been computed, 

 by means of the formulas now to be explained. But the term 

 f Q3 will afterwards be discussed, and its value investigated, 

 in order that it may be taken into account, if this should be 

 found necessary, in the progressive improvement of the theory. 

 When the term^ ' Qg is left out, the expression of the re- 

 fraction will be 



89 = sin 5 x"^^.{Qo + >^Qi-fQ,): 



and if the equivalent series be substituted for the first two terms, 



2 

 and the series for Qg be multiplied by/ = — , the result will 



. , a(l+a) r 



be 8 9 = sm S X \,r-J- >< < e 



V B t L. 



+ 0-821921 . e^ 



4- 0-552697 • e^ 



+ 0-302296 . e^ 



+ 0-132123 .c» 



4- 0-043365 . e" 



+ 0*007883 . e^^ 



— 0-002040 . e'^ 



— 0-002930 . e^'^ 



— 0-001842 . e'» 



— 0-000893 . e^^ 



— 0-000371 . e-'^ 



— 0-000139 . e'^^ 



To bring this formula to a form more convenient for calcu- 

 lation, all the coefficients must be reduced to seconds. The 

 negative terms are all very small, never amounting to so 

 much as 6'', and of no account whatever, except the apparent 

 altitude be equal to 2° or less ; it will therefore be proper to 



