MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 195 



|.= (I-y)c-«, 

 we deduce 



i«& (7) = «* + log r=^ = ^ + ^ + ^ + T + &C. ; 



and, by substituting the value of q, 



By means of this series and the value of a in terms of u already found, it is easy to 

 deduce 



and, by substituting the value of u, we finally obtain 



= log 7 X ^ • (1 - -I- ? - i^/^, -/}) ■ g^&c). . . (B.) 



<r =: 



z 



This formula determines a perpendicular ascent z, when the difference of the press- 

 ures, and of the temperatures, at its upper and lower extremities, have been found. 



The formulas that have been investigated are true in an atmosphere of air mixed 

 with aqueous vapour, as well as in one of perfectly dry air ; but in applying them, 

 perspicuity requires that the two cases be separately considered. 



Atmosphere of dry air. 

 In applying the formula (A.) to the experimental ascents that have been made in 



the atmosphere, c may be accounted equal to z, the height ascended : for — , which 



is a minute fraction at the top of the atmosphere, is insensible in small elevations. 

 Further, in such experiments, the depression of the thermometer, or the difference of 

 the temperature at the upper and lower extremities of the ascent, is only a moderate 

 number of degrees ; and as j8 is a very small fraction, the value of g in the formula 



will be so inconsiderable, that its powers may be neglected. Attending to what is 

 said, the formula (A.), even in those cases where the ascents are most considerable, 

 may take this very simple form without much error, or rather with all the accuracy 

 warranted by the nature of such experiments, viz. 



. _ ^ L±/ ^(^-"^) 

 or, by making D = g>' (1 + /3 r'), 



2 c 2 



