Terrestrial Refraction. 563 



The equivalent formula for //, involving the dip D = 7r/2 — </>, 



is 



sin/D + A^> — «o— |) 



7i = 2Ksin^ '^' T-r ~. . . (41) 



2 cos (D + A</>-g>o) 



In dealing with (41) the assumptions ordinarily made are 

 the same as before, except that (iv.) is replaced by 



(v.) my/2 tan D is negligible in comparison with unity. 



The resulting formula has the well-known form 



s=- ,._ -. . . (42) 



tan D + Y tan 2 D - — —, - 



The preceding work enables us to examine these assump- 

 tions numerically. 



Assumption (i.) is true to within 1 part in 8000 for 

 h<$ km., and cf> <90 r . 



Assumption (iii.) holds to within 1 in 400 for h < 8 km. 

 and (/>o< 90°; or to 1 in 800 for 7i<4 km. 



Assumption (iv.) is accurate to within 1 in 800 for 

 h < 8 km. and 40° < <j> < 90°. 



In assumption (v.) we may write, except when I) is 

 small, -m 7/2 cot 0o instead of w 7/2 tan D; and so (v.) 

 holds whenever (iv.) is true. 



Only (ii.) is left. This we now consider. 



Writing w = e A^ so that A</> _ w o = (l — e )A</>, we have 



W ° < 1-22 for li < 8 km. and 6 < 90° ; 

 A9- w o 



< 1-13 for h < 4 km. and O < 90°. 



A9 — ®o 



Writing further 



em/ 2 ?»/ 7 



and A , 2(1 — e)>/// 7 



A</>-wu= 6> • - 2 , 



we have the following formulae instead of (40) and (42) : 



1 , * 2 A 2eroA 

 ft = 5 cot0 o + ^g^l- —J (43) 



and , i. n ,s " /1 2(1 — e)mA 



A - stanD-^1- - j. . . (44) 



