548 Mr. A. R. McLeod on 



makes with a fixod radius, so that 



= ^dl. 



dr 



Then, since — = cos <£, )->) 



we have 



1 dty + y) _ dcj) sine/) 



/o dl ~ ~ dl. r j 



Substituting (3) and (4) in (2), we have the following well- 

 known formula, in which h — r — R is height above the earth's 

 surface : 



1 rfn 1 , 



-~77+ — r— - = (d 



/i dk p sin 



Equation (5) must yield the law of variation of refractive 

 index with height, for any assumed form of the radius of 

 curvature p. By (1) we may write (5) in the form 



~;+4=o ( 6) 



rr dh pk ' 



Since the relation giving n in terms of h cannot depend 

 upon the particular ray by means of which (6) has been 

 derived, (6) cannot depend upon k, and so pk must be inde- 

 pendent of the zenith distance at the ground, <p , of the ray 

 under consideration. Since pk is a function only of n or //, 

 we must therefore have 



>-?{/(4)y m 



where f(n, A/R) does not involve k and has no dimensions. 

 Hence (6) becomes 



1 dn r ( h \ 



The simplest assumption thatcan be made about ihe radius 

 of curvature is that it is constant for a given ray. This 

 amounts to taking only the constant term, g, in the expansion 

 ■of f(n. ///R), so that (7) then becomes 



R 2 



' = 1 (9) 



We then have 



±dn gr 



