198 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



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 alti- 

 tude. 



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 coefiicienty'. One remark respecting this quantity deserves to be noticed. By 

 expanding the formula (4.), we obtain, 



the exact value is 



YTJ^^ = 1 -A +/• 172 -/• 17273 + ^C" 



now, as these values continue very nearly equal to considerable elevations, the first 

 terms of the two series must nearly coincide : which requires that f shall be only 

 a small part off. 



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 neglected : so that. 



ff = 



1 + ^ 



a 



and, because 



2 1+/3t^ 'p'(1+/3t') ^' 



. = 0+^).l(.h-,(^-F))-1o.©. 



Now this is nothing more than the usual barometric formula for measuring heights, 

 as it is found in the writings of Laplace or Poisson. It supposes that unit repre- 

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

 force in different latitudes must be taken into account, nothing more is requisite than 

 to multiply by the proper factor. When this is done the foregoing expression will be 

 identical with the usual formula, all its minutest corrections included. But there is 

 this difference between the two cases, that the usual formula is investigated on the 

 arbitrary supposition that the temperature is constant at all the points of an elevation, 

 and equal to the mean of the temperatures at the two extremities ; whereas the other 

 expression is strictly deduced from the general properties of an atmosphere in equili- 

 brium. The exact theoretical formula has been made to coincide with the approxi- 

 mate one, by dismissing all the terms that cannot be estimated in the present state 

 of our knowledge of the phenomena of the atmosphere. 



All the properties of the atmosphere that have been ascertained with any degree of 

 certainty, have been made known to us by the application of the barometric formula: 



