Mr. Ivory 07i the Theory of the Astronomical Refractions. 5 



verified at the earth's surface, it follows that the supplement- 

 ary function <^ [u) will vary slowly as u increases, that is, as 

 the density of the air decreases in ascending. This proves that 

 the approximate equation (4.) will be very little different from 

 the exact equation (5.) at great elevations in the atmosphere. 



At the surface of the earth d u is equal to the variation of 

 the density for a depression of the thermometer expressed by 

 dT'. and although the proportion of these two quantities can- 

 not be ascertained by direct experiment, yet, as is shown in 

 the paper of 1823, it is easily deduced from the rate at which 

 the temperature decreases as the height increases, which rate 

 is easily determined experimentally. The quantity f thus 

 found is necessarily constant at the same observatory. It is 

 the mean of all the occasional values, which vary incessantly, 

 while the real atmosphere undergoes every vicissitude of which 

 it is susceptible. The mean refraction andy are invariable in 

 quantity, because they depend alike upon the mean condition 

 of the air at a given place. Some confusion has arisen on 

 this point from not distinguishing between the mean refrac- 

 tion of a star and its true refraction in a variable atmosphere. 



In all that has been said there is no supposition of an ar- 

 bitrary constitution of the atmosphere. The assumed formula 

 (4'.) is an approximate truth in every invariable state of the 

 air in equilibrium. Conceive a cylindrical column of air 

 perpendicular to the earth's surface, and extending to the top 

 of the atmosphere; at the height where the temperature is 

 T, and the density §, let jp denote the weight of the column 

 above the height ; and suppose that "p^ g, t are changed into 

 p', g', T at the surface of the earth ; because the pressures are 

 proportional to the elasticities, we have the usual equation, 



or, which is the same, 



and by substituting the complete expression of the tempera- 

 ture as given in (5.), we shall obtain, 



i^ = c-«-/(c-"--c-2«)-c-«x <p (?/): (6.) 



and if we omit the supplemental part, which is small even at 

 great elevations, the result will be, 



|, = c-«-/(c-»-c-2«). 



