418 Count Paul de Saint-Robert on Atmospheric Refraction. 

 refraction, we shall have 



1 

 cosv=l — 



(l-«flr )(l+«r ) 



shW= r^-y/\+Up . 



Taking 



we get 

 whence 



l + «r 



r = 20,888,782 feet, 

 a= 000002276, 

 4^ =0-00058856, 



v=4°0'12", */=86° 33' 25", 



R = 33' 37", 



a very satisfactory value. It is often repeated that the refraction 

 computed on the hypothesis of uniformly decreasing density is 

 less than the truth. But I may observe that this happens only 

 when the rate of the decrease of density is taken too small. All 

 works which treat of astronomical refraction on the assumption 

 of an equable decrease of density take the height of the atmo- 

 sphere at double that of a homogeneous atmosphere. Now 

 although this height is that derived from the condition that the 

 pressure and density are reduced to zero simultaneously at the 

 upper limit of the atmosphere, yet it does not correspond to the 

 real decrease of density observed in the inferior strata. The 

 consideration of the higher strata of the atmosphere, which have 

 no sensible effect on refraction, is of little importance to us. 

 What we want is the rate of decrease of the density in the infe- 

 rior portion of the atmosphere ; and this experience shows us to 

 be larger than that derived from an atmosphere of double the 

 height of a homogeneous atmosphere. 



For the purpose of the measurement of heights, we derive 

 easily, from the preceding expression of the radius-vector, that 



^(r-r )=^ = 2r Csin|sin^V-|J, 



which, added to the expression 



r—r n =%-\ } 



will give the difference of level of two stations, when the 



zenith-distance 6 at one, and the angle v at the centre of the 



earth, between the radii drawn through the stations, are known. 



The radius r being thus determined, the difference between 



