( 585 ) 



r= r + h, 



(i index of refraction ;it the surface of the earth, 



ft „ „ „ „ „ height h, 



q density of' the air al the surface of the earth, 



o density at the height h, 



t temperature at the surface of the earth, 



/„ height of a column of air of uniform density at 45° latitude, 

 of a temperature t , which will he in equilibrium with the pressure 

 of one atmosphere, the gravity being the same at different heights. 

 According to Regnault's constants, we have /„ = 7993 (1 -j- at ) 

 meter, if a represents the coefficient of expansion of the air. 



Between these quantities exist the following relations: 



li 2 = 1 ~f- 2 co (c being a constant), id = 1 — - 

 co n a R lih 



e = — a y ■= 



l+2co sinl" l J (r + h)l 



To determine the value of (h at each height, we require a relation 

 between to and y or between to and h, which can be obtained when 

 we assume that the temperature varies according to Ivory's theory, 

 or that the temperature varies as represented in table IT. For the 

 same given values of z and co, the two values of' r/.s' in formula (I) 

 can be computed by means of' the first and by means of the second 

 supposition, and the ditferences of' these two values of ds can be 

 found. By means of' mechanical quadrature, we can then determine 

 the ditferences As of' the refractions s according to Ivory's theory 

 and according to table II. 



The relations between y and to may be found in the following 

 manner. 



5. If in a given horizontal initial plane, at a distance r e from the 

 centre of the earth, the pressure is p , the temperature t and the 

 density of the air o , and in another horizontal plane, h kil. above the 

 former, the pressure is p, the temperature t, the distance from the 

 centre of the earth r, and the density of the air o, then we have 

 (see Radau) : 



o Eh 



or, putting - = r t and = y : 



