Mr. Ivoiy on the Theot'y of the Astronomical Refractions. 151 



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 publication of 

 the Tabula Regiomontance, he found that the table agreed with 

 Bessel's observed refractions to the distance of 88° from the zenith, 

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

 observations 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 this formula being to ex- 

 press the temperature in terms of such a form as will produce, in the 

 refraction, independent parts that decrease rapidly. By this means 

 he proceeds in the analytical investigation of the problem in its more 

 comprehensive form, and deduces two equations on which its solu- 

 tion depends. 



The first of these contains the law according to which the heat 

 decreases as the height above the earth's surface increases ; and the 

 second determines the perpendicular ascent, when the difference of 

 the pressures and of the temperatures at its upper and lower extre- 

 mity have been found. If the latter, with a slight transformation, 

 be multiplied by the proper factor, representing the variable force of 

 gravity in different latitudes, it becomes identical with the usual 

 barometric formula, all its minutest corrections included; and it has 

 this advantage ; that, whereas the usual formula is investigated on 

 the arbitrary assumption, that the temperature is constant at all the 

 points of an elevation, and equal to the mean of the temperatures at 

 the two extremities, this formula is strictly deduced from the gene- 

 ral properties of an atmosphere in equilibrium. 



Having determined, from experimental results, the values of cer- 

 tain constants in these formulae, — first, in an atmosphere of dry air, 

 and, secondly, in an atmosphere of air mixed with aqueous vapour, 

 the author remarks, that the analytical theory agrees in every re- 

 spect with the real properties of the atmosphere, as far as these have 

 been ascertained. 



The object of Mr. Ivory's further investigation is to show, that 

 the same theory represents the astronomical refractions with a 

 fidelity that can be deemed imperfect only as far as the values of 

 particular constants, which can only be determined by experiment, 

 are liable to the charge of inaccuracy. He therefore proceeds to 

 determine, from the formulae previously deduced, the refraction of a 

 star in terms of its apparent zenith distance. For this purpose, the 

 differential equations are transformed by the introduction of new 

 symbols ; the limits of certain terms are determined previously to 



