LIMIT OF THE ATMOSPHERE KERRER 53 



Let a and a' be the conjugate points; D and D' their distances 

 from the center of the earth ; a and a' the angles of divergence of 

 the corresponding pencils; then in the neighborhood of the zenith 

 and according to the theorem just stated we have 



D .« = n' (16) 



D' a! 



But from the triangle mca' since y = £ there results 



D' r 



Z- = y or D' a' = R r 

 R a' 



substituting this in equation (16) gives us 



D a 



R'r 



(17) 



Furthermore from the triangle mad, designating the altitude of the 

 zenith point by h we have 



a R + h 



C +P D 



or 



D a = (R + h) (c + p) 



so that equation (17) becomes 



i+ i)KH (i8) 



If in this we substitute for -its value from equation (2) we get 



r 



h = [(n' - 1) - 57.3* »' (1 + J)] R = about .027 kilometers. (19) 



For zenith distances up to i° the altitude of the zenith point is 

 independent of p and £; or, for the same locality, R, and the same 

 condition of the atmosphere, »', the zenith point has an invariable 

 position. 



