MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 191 



Astronomers are in the habit of using different tables or formulas of refraction, 

 which, being derived from conjectural views, do not agree with one another, except to 

 a limited distance from the zenith. Now this is contrary to the vei*y conception we 

 have of the mean refractions, which are determinate and invariable numbers, at least 

 at the same observatory. 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 produce the phenomena. Such a table adapted to every 

 observatory, 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 re- 

 spect : for if the mean refractions were accurately settled, the uncertainty in the place 

 of a star would fall upon the occasional corrections depending on the indications of 

 the meteorological 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 sepa- 

 rated 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 follows 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 pur- 

 pose it will be requisite to employ certain functions of a particular kind, viz. 



Ri=l-c-", 



R2 = 1 — w — c~ **, 



R3 = 1 — W -f. p-^ — C *, 



K^ _ ^1 u-i-^,^ l.2.3**^ 1.2.3. ..«- 1/ ^ • 



In these expressions c is the number of which the hyperbolic logarithm is unit ; and 

 it is obvious that R^ is zero when m = 0. These expressions have several remarkable 

 properties, which are proved by merely performing the operations indicated. 

 1st. d.R^_ 



J — Ridu = Ri^i, 



the integral being taken equal to zero, when w = 0. 

 2ndly. rf.c-"R. 



