4- Mr. Ivory on the Theory of the Astronomical Refractiotis. 



template an atmosphere fixed in its condition at any given 

 place or observatory, being supposed a mean between all the 

 variations that actually take place in an indefinite time. In 

 such an atmosphere the temperature and pressure at the 

 earth's surface will be mean quantities deduced from obser- 

 vation : the air at all elevations will have an elastic force equal 

 to the incumbent weight which it supports, as an equilibrium 

 requires : and, whether the air be dry or moist, its refractive 

 power will be equal to the refractive power of dry air sub- 

 jected to the same pressure and temperature *. These pro- 

 perties of the mean atmosphere rest upon experiment and de- 

 monstration: in other respects its nature is not directly 

 known to us : and the laws of its action can only be discover- 

 ed, not by hypothesis, but by observation. 



The consideration of a mean atmosphere, invariable at any 

 given observatory, is a necessary consequence of the notion 

 we attach to the mean refractions ; for these would be realized 

 in such an atmosphere: but they are different in any other 

 state of the air. 



These observations being premised, if the formula (4.) be 

 verified at the earth's surface in any invariable atmosphere, 

 by giving a proper value to the constant^^ it will still hold, at 

 least witli a very small deviation from exactness, at a great 

 elevation, probably at a greater elevation than has ever been 

 reached by man. In order to prove this, let the arbitrary 

 function <$ {u) be added, so as to complete the formula by 

 rendering it perfectly exact : then 



|±^, =: 1-/(1 -C-)-M"), (5.) 



and it will follow that 45 {») = 0, when u = 0, that is, at the 

 earth's surface. Again, differentiate the equation, observing 

 that T decreases when u increases, then 



/3 dr _ d . <p {u) ^ 



1+/3t/ du -^ '^ du ' 

 now, since this equation is true for all values of w, it will liold 

 at the earth's surface, or when u =: 0: and if/ be taken equal 

 to the particular value of 



<3 du 

 1+/3t" dr ' 



when 2t = 0, it will follow that — LzA_i — q yvhen u = 0. 



d u 



And since the equations <b (n) = and ^_L^_v!fl _ q Q,.g ij^|.]j 



du 



* Additions a la Conn, des TevtjJs, 1839, p. 36. 



