63 



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 

 gra\aty 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. 



Ha\ing 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, 

 pje 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 

 their being neglected ; and the equation is finally reduced to a form, 

 in which the remaining operations consist in investigating the inte- 

 grals of four expressions, and in subsequently assigning their nume- 

 rical values. Great skill is displayed in conducting these intricate 

 investigations ; and after going through the most laborious calcula- 

 tions and computations, the author exhibits a table of theoretical re- 

 fractions, deduced solely from the phenomena of the atmosphere, 

 for zenith distances, extending from lO'^ to 89 These refractions 

 are compared with those in Bessel's table, in the Tabula Regiomon- 

 tance, and also with those in the table in the Connaissance des Temps, 

 From this comparison, it appears, that the three tables agree within 

 less than 1", as far as 80° from the zenith: from 80° to 88° of zenith 

 distance, the numbers in the French table exceed those in Bessel's, 

 the excess being 2" at 84°, and 4" at 88° ; and with a single excep- 

 tion at 88°, (probably, judging from the character of the adjacent 

 number, arising from an error of computation,) the refractions in 

 the new table are nearer to Bessel's than those in the French table ; 



