158 BELL SYSTEM TECHNICAL JOURNAL 



integrating which from 6 = Q to 6 we obtain 



t - to ^ ro_cosj/' ^^^^ ^^^ - V') + tan ,A]. (9) 



If 5 in (5) is made somewhat less than one in absolute value and is 

 negative, we obtain a fairly good approximation of the distribution 

 actually encountered. The exponent 5+1 has then a small positive 

 value and from (5) 



1 r 



(10) 



dnjdr n{s + 1) 



Since n is very close to unity and since we may assume p/r = 4.0 

 (Appendix II), we find that 5 = — 0.75. This value will be used 

 later. 



Consider now a second series of values in equation (9), /', ro', s', etc. 

 which represent another situation which we shall define as follows: 



5' = — 1 (i.e., constant index and no bending of the rays) 

 Tp' = \p {i.e., no change in the initial direction of the ray) 



s'9' = sd and ro' = --, 

 s 



so that ro'd' = rod, that is, the peripheral distance traveled is the same 

 in the two cases although the radius of the earth has been increased 

 from ro to ro{— l/s). 



By substituting these new primed values for the unprimed values in 

 equation (9), we obtain 



f/ _ ^„/ = ^° ^os ^ _ ^^^ ^^ ^^j^ 



cs 



which is identical with (/ — to) in (9). Note that the only assumption 

 that has been made, limiting the generality of this equivalence, is the 

 special distribution assumed in (5). The phase time is therefore un- 

 altered by this substitution. 



We have yet to prove, however, that in these two cases, rays leaving 

 at the same angle, (^p = yp'), and describing angles d and 6' at the real 

 and fictitious centers of the earth, will have the same increase in 

 elevation above sea level. If this can be shown, the equivalence will 

 have been completely established. 



The increases in elevation of the ray above that of the starting point 



