Mr. Ivory on the T/ieoty oj'the Asiroiio/jiical lief r act ions. 319 



air; its power of absorbing heat; and the interchange by ra- 

 tliation which takes place with the eartli, with the etherial 

 spaces above, and with the stars visible above the horizon. 

 So many conditions, placed beyond the reach of our inquiry, 

 may well puzzle the most expert algebraist to take them into 

 account. But it may be doubted whether this be the proper 

 view of the problem. The astronomical refractions at any 

 observatory are mean effects of the atmosphere ; and it may 

 be alle"ed that the proper way of accounting for them is to 

 compare them with other mean effects of the atmosphere at 

 the same place. 



4. In 1715, twenty years after the researches of Newton, 

 Brook Taylor published in his Methodus Incremetitorum, the 

 first investigation of the astronomical refractions on the sup- 

 position that the density of the air is variable. The diff(:;r- 

 ential equation is accurately deduced from the principles laid 

 down by Newton ; which removed all the difficulties of the 

 problem in this respect. 



Kramp, in his Analj/se des Refractions Astronomiques et 

 Terrestres, has elucidated the elementary parts of the pro- 

 blem, and has greatly improved the mathematical solution. His 

 method is particularly commodious and useful in the case of 

 the horizontal refraction. For altitudes above the horizon 

 the integrals are not susceptible of being simply expressed, 

 and seem to require the aid of subsidiary tables in applying 

 them. 



The problem of the refractions being an important one in 

 astronomy, many solutions of it have been published by dif- 

 ferent geometers. Some of these are preferable to others, 

 because the method of calculation is easier in practice. For 

 altitudes greater than \{f, they may all be reckoned equiva- 

 lent, differing from one another only in the quantity assumed 

 for the refractive power of air. They also mostly agree in 

 the horizontal refraction, which is taken from observation. 

 But for altitudes less than 16^, they are different ; because, at 

 these low altitudes, the refractions are affected by the arbi- 

 trary suppositions used in constructing the tables. 



'J'homas Simpson published a judicious dissertation on this 

 problem. He distinctly points out that, to a considerable di- 

 stance from the zenith, the refractions are indepentlent of the 

 manner in which the atmosphere is supposed to be consti- 

 tuted. In comparing the two atmospheres that have densities 

 decreasing in arithmetical and geometrical progression, he 

 remarks that the horizontal refraction comes much nearer the 

 observed quantity in the first atnjospiiere than it does in the 

 second; for which reason he gives the prcleroiice to the first 



