188 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



surface in any invariable atmosphere, by giving a proper value to the constant /, it 

 will still hold, at least with a very small deviation from exactness, at a great eleva- 

 tion, probably at a greater elevation than has ever been reached by man. In order 

 to prove this, let the arbitrary function (p (m) be added, so as to complete the formula 

 by rendering it perfectly exact : then 



[±|^= 1-/(1 -c- »)-?(«), (5.) 



and it will follow that (p (u) = 0, when u = 0, that is, at the earth's surface. Again, 

 differentiate the equation, observing that r decreases when u increases, then 



now, since this equation is true for all values of u, it will hold at the earth's surface, 

 or when u = 0: and iff be taken equal to the particular value of 



/3 du 



l+^T''dT 



when 2^ = 0, it will follow that '^^ = 0, when u = 0. And since the equations 



<p (u) = and 'J^ = 0, are both verified at the earth's surface, it follow^s that the 



supplementary function (p {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 du is equal to the variation of the density for a depres- 

 sion of the thermometer expressed hy dr: and although the proportion of these two 

 quantities cannot 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 tiie 

 height increases, which rate is easily determined experimentally. The quantity y 

 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 and/" 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 refraction of a star and its true refraction in a variable atmosphere. 



In all that has been said there is no supposition of an arbitrary 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 r, and the density §, let p denote the weight of the column above 

 the height ; and suppose tljat p, f, r are changed into p', §', r' at the surface of the 



