108 Mr. Ivory on the Theory of the Astronomical Refractions. 



accounts sufficiently for the excess of :' deduced from 



Gay-Lussac's ascent, above the other values found from mo- 

 derate elevations. Without further research we may adopt 

 the following determinations as near approximations derived 

 from a multitude of experiments, 



The difference of these numbers from those used in the paper 

 of 1823, produces an increase in the refractions amounting 

 to 19" at the horizon, and to 2" at 2° of altitude. 



The irregular manner in which the heat decreases in such 

 experiments as have been used for finding f evidently makes 

 them altogether unfit for determining the next coefficienty'. 

 One remark respecting this quantity deserves to be noticed. 

 By expanding the formula (4.), we obtain, 



the exact value is 



TTF' = ' -•^" + (/--^ ) • TF - «"^- = 



now, as these values continue very nearly equal to consider- 

 able elevations, the first terms of the two series must nearly 

 coincide : which requires that f shall be only a small part 

 of/ 



We have next to attend to the formula (B). As q is only 

 a small fraction in all the elevations that have been reached 

 in the atmosphere, its square and other powers may be neg- 

 lected : so that, 



= 4 ('-!-) log (^) 



a 



and, because 



r' + T, 



1 _ i = —lA-Xj .___^__ _ T . 



2 14-^t' '§'(1 + ^x0 



= (.+ £-). L(, + ,3(i:f^)).,og(£'). 



Now this is nothing more than the usual barometric formula 

 for measuring heights, as it is found in the writings of La- 

 place or Poisson. It supposes that unit represents the force 

 of gravity at the earth's surface; and if the variable intensity 



