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° from 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-
