(i/j- i^i') - (f/2) cos^a^ [(1^2- i^i') - sin (i/^'- i^/ ) cos (i/,'+ t'/)] 



+ f^ 5^(1^2'- v^Y cot (v^- Vi) cos^a„- %{v2- v'lV cotd/j'- i^,') cos''ao 

 ~%{v2- 1^1')^ cscCi/j'- i^j') cos^Oq cos(i/i'+ 1^2') 

 + %{v2 — ViY 030(1/2'" "^i') oos''a(, COS {vi + V2) 

 — (1/16) sin 2 (i/j'- i^i') cos ''oo cos^(i/,'+ i/j') 

 + (1/16) (1^2'- 1/,') cos^ao + (1/32) sin 2(1^2'- h') cos''ao 



Now in (129) let Cq = 90°- 6^, d, = v^, d2 = i^j' , d = dj - dj = i/j' - I'l' and the equation 



(129) 



— = d - (f/2) [d sin '00 - sin d sin '0ocos(di + dj) ] 

 a 



+ Y 



(130) 



M d' cot d sin'^o - M A^ cot d sin "0, 

 - ^ d' esc d sin '^o cos (dj + d2) 

 + 5i d' osc d sin "^o cos (d, + dj) 

 -(1/16) sin 2d sin X cos'(di + dj) +(1/16) d sin "^o +(1/32) sin 2d sin "^o _ 



Since 0^ is the parametric latitude of the vertex of the Great elliptic arc, we have ( or 

 may place) 



(sin ^1 + sin 6^^ (sin 0, - sin 0-^ 



X = 



1 + cos d 1 - cos d 



(sin 0, + sin d^^ (sin 0^ - sin d^) ' 



1 + cos d 



1 -cos d 



2 sin '00 , 

 -■ 2 sin '00 cos (d, + d2) 



(131) 



From (131) sin'0o = X/2, sin '0o cos (d, + dj) = Y/2, and we can write (130) in the form: 

 — = d - (f/4) (Xd - Y sin d) 



+ (f7l28) 



(132) 



(16d' cotd)X -(16d' cso d) Y 

 + (2d + sin 2d - 8d' cot d) X' 

 _+ (8d' cscd) XY - (2 sin 2d) Y' 

 If we factor sin d out of every term of (132), we can write: 



S = a sin d [T - (f/4) (TX - Y) + (fV64) (A^X + B^Y + C„X' + DpXY + E„Y')] 



T = d/sin d, E„ = -2 cos d, Ao = - DoEo, Q = T - '/^(Ao +£„), (133) 



Do = 4T', Bo = -2 Do, d is the spherical distance between the points Pi(0i,Ai) and P2(02,Aj) 

 given by some spherical formula as 



cos d = sin 6^ sin 02 + cos 0, cos 02 cos AA, AA = A2 - A,. 



86 



