8 Mr. Ivory on tJie Theory of the Astronomical Refractions. 



formulas of refraction, which, being derived from conjectural 

 views, do not agree with one another, except to a limited di- 

 stance from the zenith. Now this is contrary to the very 

 conception we have of the mean refractions, which are deter- 

 minate and invariable numbers, at least at the same observa- 

 tory. A great advantage would therefore ensue from setting 

 aside every uncertain table, and substituting in its place one 

 deduced from the causes really existing in nature that pro- 

 duce the phaenomena. Such a table adapted to every obser- 

 vatory, if this were found necessary, would contribute to the 

 advancement of astronomy by rendering the observations 

 made at different places more accurately comparable. It 

 might contribute to the advancement of knowledge in another 

 respect: for if the mean refractions were accurately settled, 

 the uncertainty in the place of a star would fall upon the oc- 

 casional corrections depending on the indications of the me- 

 teorological instruments ; and it is not unreasonable to expect 

 that much which is at present obscure and perplexing on this 

 head might be cleared up, if it were separated from all foreign 

 irregularities, and made the subject of the undivided attention 

 of observers. 



7. The paper in the Philosophical Transactions for 1823 

 takes into account only the rate at which the densities in a 

 mean atmosphere vary at the surface of the earth ; what fol- 

 lows is an attempt to complete the solution of the problem 

 by estimating the effect, of all the quantities on which the 

 density at any height depends. For this purpose it will be 

 requisite to employ certain functions of a particular kind, viz. 



Rj = 1 - c-«, 



R^ = 1 _ ?< — c-« 



R. = 1 - « 4 



1.2 



In these expressions c is the number of which the hyperbolic 

 logarithm is unit; and it is obvious that R,- is zero when 

 u = 0. These expressions have several remarkable pro- 

 perties, which are proved by merely performing the opera- 

 tions indicated. 



1st. d.lli _ 



