62 



in which p stands for the dilatation of air or gas by heat, r' and r" 

 for the temperature at the earth's surface, and at any height above 

 it, and c"" for the density of the ah- at that height' in parts of its 

 density at the surface. If this formula be verified at the earth's sur- 

 face in any invariable atmosphere, by giving a proper value to the 

 constant /, it will still hold, at least with a ver^- small deviation from 

 exactness, at a great elevation; and this is immediately shown. 



This manner of aming at the constitution of the atmosphere is 

 contrasted with the procedure of M. Biot of transforming an alge- 

 braical formula, for the express purpose of bringing out a given re- 

 sult. As the problem in the Mecanique Celeste is solved by means 

 of an interpolated atmosphere between two others; as in Mr. Ivory's 

 paper of 1823, there is no allusion to such an atmosphere; and as 

 the table in that paper is essentially different from all the tables 

 computed by other methods, he contends that all these must be suf- 

 ficient to stamp an appropriate character on his solution of the pro- 

 blem. But if ingenuity could trace some relation, in respect of the 

 algebraic expression, between the paper of 1823 and La Place's cal- 

 culations, he considers that it is not difficult to find, between the 

 same paper and the view of the problem taken by the author of the 

 Principia in 1696, an analogy much more simple and striking. 

 Newton having solved the problem, on the supposition that the den- 

 sity of the air is produced solely by pressure, and having found that 

 the refractions thus obtained greatly exceeded the observed quantities 

 near the horizon, inferred, in the true spirit of research, that there 

 must be some cause not taken into account, such as the agency of 

 heat, which should produce, in the lower part of the atmosphere, the 

 proper degree of rarefaction necessary to reconcile the theoretical 

 with the observed refractions. The author's sole intention, in intro- 

 ducing the quantity/ in his formula, is to cause the heat at the 

 earth's surface to decrease in ascending, at the same rate that ac- 

 tually obtains in nature, not before noticed by any geometer, but 

 W'hich evidently has the effect of supplying the desideratum of 

 Newton. 



The author considers, that the comparison of the table in the pa- 

 per of 1823, with the best observations that could be procured at the 

 time of publication, was satisfactory ; and after the pubhcation of 

 the Tabula Regiomo?ita?ia, he found that the table agreed with 

 Bessel's obser^^ed refractions to the distance of 88° fi'om the zenith, 

 with such small discrepancies as may be supposed to exist in the 

 obsen-ations themselves. 



The paper in the Philosophical Transactions for 1823, however, 

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

 atmosphere, vary at the surface of the earth ; but, in the present 

 communication, the author proposes to effect the complete solution 

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

 the density at any height depends. For this purpose, he finds it ne- 

 cessary to employ functions of a particular kind ; and then gives a 

 formula, one part of which consists of a series of these functions, for 

 the complete expression of the temperature of an atmosphere in 

 equilibrium ; the intention of assuming thk formula being to ex- 



