(These were apparent from inspection of Figure 13 but illustrate the use of (6)). 



From Figure 13, the tangent t2 to the elliptic section lying in the plane (5) is the line of 

 intersection of the planes (4) and (5). From (4) and (5) we have respectively Aj = N, cos (;6, , 

 Bj = 9, Ci = Nj sin 0, ; Aj = A, Bj = 6,0^= -C and applying (6) find the direction cosines of 

 tj to be 



Pj = (-B sin 0i)/d, qj = (A sin <^i + C cos 0i)/d, Fj = (B cos 0,)/d 

 where d = [ B' + (A sin <;6, + C cos 0^ Y ] '/'. (8) 



The forward azimuth OAg from Qi to Q^, as shown in Figure 13, is the angle reckoned 



clockwise from south between the tangents tj and t^. Hence from (7) and (8) 



B . 2 I B 2 B , . 



cos ay^g = p,p2 + qiq^ + t^t^ = - — sin <p^ cos <pi = - , (9) 



d d d 



d - [ B' + (A sin 0, + C cos cf>,r] '/' 

 Since cot a^g = cos a\^/{l - cos^ay^g) '^^ we have from (9) that 



cot a^g = - B/(d' - B^) '/^ (10) 



Now d' -B' = B' +(A sin 0, + C cos 9!)i)'-B' = (A sin (f),+ C cos <f) y, 

 so \/d^ - B^ = A sin 0, + C cos 01 and (10) may be written 



cot ay^g = - B/(A sin 0, + C cos 0i). (11) 



With the values of A, B, C from (5), equation (11) may be written as 



[sin 02 — (N,/N2) sin 10 ] e ^ cos j sec 02+ (sin j cos A A — tan 02 cos 0i). 



c°t«AB= : (12) 



sin A A 



Referring again to figure 13, it is seen that from considerations of symmetry, we have only 



to interchange the subscripts 1 and 2 and change A A to - A A in (12) to obtain cotg^ (the back 



azimuth on the other normal section). We thus obtain from (12) 



[sin 01 - (N2/N1) sin 02]e^ cos 02 sec 1 + (sin 2 cos A A - tan jCos 02) , . 



cot apjA = : T-; ^'-■^) 



o^ sin AA 



GREAT ELLIPTIC SECTION AZIMUTHS 



Figure 14 shows the great elliptic section and azimuths as abstracted from Figure 12. The 

 same coordinate system is used as in Figure 13 so that most of the equations developed with the 

 normal section azimuths can be used. The angle oy^^g between the tangents ti and tj is the 

 forward azimuth required. We already have the direction cosines of t, see equations (7). The 

 tangent t2 is the intersection of the great elliptic plane with the tangent plane at Qi, equation (4). 

 The equation of the great elliptic plane through Qj, Q2, using equations (1), is given by the determinant 



45 



