MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 201 



we shall obtain the following equation, which is sufficient for the problem of the re- 

 fractions in an atmosphere of moist air : 



o— 5 — i:\'-r'\ u -/• -u^ —f -u^ 2 ^ — &c. ;► . . . (10.) 



In which expression the coefficients/,/', &c., may be considered the same in all at- 

 mospheres, the quantity u varying from zero at the earth's surface to be infinitely 

 great at the top of the atmosphere. 



8. In the foregoing analysis, every formula has been strictly deduced from the equa- 

 tions 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 phenomena of the atmosphere with the requisite accuracy. It may not be im- 

 proper to notice here an obvious consequence of the equation 



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



being extended from the surface of the earth to the top of the atmosphere, is the ana- 



lytical 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 to py : 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 /,/', &c., which can only be deter- 

 mined by experiment, are liable to the charge of inaccuracy. 



9. The apparent zenith-distance of a star being represented by 6, and the refraction 

 by ^ ^, the following formulas have already been obtained (^ 2. equations (2.) and (3.)), 



MDCCCXXXVIII. 2 D 



