888 Mr. Ivory on the Theory of the Astronomical Refractions. 



8. In the foregoing analysis, every formula has been strictly 

 deduced from the equations of equilibrium : no quantities 

 have been introduced except such as really exist in nature, 

 and might be determined experimentally, if we had the means 

 of exploring the phsenomena of the atmosphere with the re- 

 quisite accuracy. It may not be improper to notice here an 

 obvious consequence of the equation 



P = ?' f—d<T c-«, 

 which holds in an atmosphere of dry air ; namely, that the 

 integral f^d<TC-% 



being extended from the surface of the earth to the top of the 



atmosphere, is the analytical expression of ^, or of the 



height of the homogeneous atmosphere, that is, of a column 

 of air equiponderant to the whole atmosphere, and every part 

 of which has the same density and the same weight which it 

 would have at the surface of the earth. This height varies 

 only with the temperature, and is thus determined : 



In like manner, in an atmosphere of air mixed with aqueous 

 vapour, the same integral is equal tOy-77: and we have 



Thus the analytical theory agrees in every respect with the 

 real properties of the atmosphere, as far as these have been 

 ascertained ; and we now proceed to show that the same 

 theory represents the astronomical refractions with a fidelity 

 that can be deemed imperfect only in so far as the constants 

 ff'i &c., which can only be determined by experiment, are 

 liable to the charge of inaccuracy. 



9. The apparent zenith-distance of a star being represented 

 by 5, and the refraction by 8 5, the following formulas have 

 already been obtained (§ 2. equations (2.) and (3.)). 



dy 



d.dQ 



V r^— 



y 



v= a sin 9 x\/i±lilll 



the (quantity 8 fl being supposed to increase from the surface 



