18 



BELL SYSTEM TECHNICAL JOURNAL 



Subscript h denotes receiving station, 



Subscript x denotes intermediate point, 



Loab = Difference in longitude between a and b. 



By the law of cosines, in a spherical triangle of sides abc and angles 

 ABC, if we are given two sides and the included angle, the side opposite 

 the given angle may be computed from (1) below. 



(1) cos a = cos b cos c -\- s\n b sin c cos A 

 or substituting geographical coordinates 



(2) cos Dab = sin <pa sin (pb + cos <pa cos ^ cos Loab- 



This may be made more convenient for logarithmic computation by 

 writing it in the following form: 



(3) hav Dab = hav ((pa — <pb) + cos tpa cos (pb hav Loab- 

 Now by the law of sines 



(4) 



sm 



sm 



^ cos (fb sm Loai, . J r^ 



Ca = -■ ?S = cos ipb sm Loab CSC Dab, 



sm Dab 

 „ COS <pa sin Loajb ■ T 7-> 



Cb = --T =i = cos ^a sm Loab CSC DoA. 



sm Vab 



Equations (3) and (4) above determine the great circle distance 

 between "a" and "6," the azimuth of "a" from "6," and "6" from 

 "a." To find the position of a point "x" located a fraction of the 

 total distance between "a" and "b" we again substitute in (1), 

 obtaining 



sin (px = sin <pb cos D^t, + cos <^6 sin Pj* cos G,, 

 + cos (Pa sin Z?ia cos Ca, 



, . fsin ^^ = sin <pb cos Z) 



[sin <pi = sin ipa cos Z>, 



and by the law of sines 

 sin Loax = 

 sin Lobj. = 



(6) 



sin Ca sin D^a 

 cos ^i 



sin Cfc sin Z)^, 



cos (ji?! 



= sin Ca sin Pia sec (px, 

 = sin Cb sin D^b sec (jsx. 



The latitude and longitude of intermediate point "x" are therefore 

 determined by equations (5) and (6) above. 



