172 



MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



from certain mean effects produced by the atmosphere at the observatory. To trace 

 the relations that necessarily subsist between the mean effects that take place at a 

 f^iven point on the surface of the earthy is the proper business of geometry : if this 

 can be successfully accomplished, the astronomical refractions will be made to de- 

 pend upon a small number of quantities really existing in nature, and which can be 

 determined, either directly or indirectly, by actual observation. 



1. The foundation of the theory of the astronomical refractions was laid by Domi- 

 nique Cassini. The earth being supposed a perfect sphere, he conceived that it was 

 environed by a spherical stratum of air uniform in its density from the bottom to the 

 top. By these assumptions the computation of the refractions is reduced to a pro- 

 blem of the elementary geometry requiring only that there be known the height of 

 the homogeneous atmosphere, and the refractive power of air. Let the light of a star 



S fall upon the atmosphere at B, from which point 

 it is refracted to the eye of an observer at O on 

 the earth's surface DOE: the centre of the earth 

 being at C, draw the radii C O K, C D B H : the 

 angle K O B = ^, is the apparent zenith distance 

 of the star ; and O B C = <p is the angle in which 

 the light of the star is refracted on entering the 

 atmosphere : now from the triangle O B C we de- 

 duce 



CO 

 sin O B C = sin K O B X ^^ * 



,. . . DB 



or, which is the same thing, putting i = -q^* 



sin d 



Again, (p being the angle in which the light of the star is refracted, if we put ^ d for the 

 refraction, the angle of incidence S B H, which in the present case is always greater 



than the angle of refraction, will he = p -]- I 0; and ^—zrirz — - will be a constant 



s *■ 



sin f 



ratio represented by . ; so that 



sin(. + S«) = ,7& = 7r 



sind 



{I + i) Vl —"zu 



Tlius we have the two following equations, which furnish a very easy rule for com- 

 puting the mean refractions according to Cassini's method, viz. 



sin . fl 



Sin <p = 



I + t 

 sin (9 + S ^) : 



sm 



(1+0 -/l -2« 



