v'+ c [(1/8) sin 2i/'- A cot i/'- — (1 + 2 tan ''ao) ] 



4 



(123) 



*P sin Uu - y? fii' cot ^Oo cot i^'h (sin 2i/'- 2v') 



4 



+ (1/256) [8 sin 2i''(l - 3 tan'a„) - sin 4i/'] + (3/64) v'iA tan'oo - D 



Referring (123) to the end points of the geodesic arc we have: 



S , , 



— = (1^2'" I'l') + c [(1/8) (sin 21^/- sin 21^5') - A (cot v^- cot i/^') - ^(1^2'- 1^ 1 ) ( 1 + 2 tan ^Oq)] 



-H A^ cot ^tto (cot 1/2'- cot 1/,') + — [(sin 21/2'- sin 2i/j' ) - 2(i^2~ i^i) 1 



4 



+ (1/256) [8 (1 - 3 tan^oo) (sin 2^2'- sin 2vi) - (sin 41/2'- sin 4i/,')] 

 + (3/64) (1/2'- i^i') (4 tan ="00 - 1) 



Note that the term W sin a^ vanishes in (124). 

 From lilt we have from the expression forA that: 

 tan Oo 



(124) 



— A (cot v'2 — cot Vi) = 



iv2- 1^1'), 



(125) 



A = %{v2- 1^1') tan^ao[cot (v^ ~ I'l') ~ esc (i/j'- i/^') cos {v^ + i/j')] 

 We list also for reference the identities: 



sin Ivi — sin 2i/i'= 2 sin (vj'— 1^1') cos (i/,'+ 1/2') , (126) 



sin 41^2' ~ sin 'iv( = 2 sin 2(1^2'— 1^1') [2 cos ^(1^1'+ i/j') — l] 

 Applying (125) and (126) to (124) we obtain: 



— = (1^2'- ^i')-(c/4) [(1/2'- 1^1') -sin (1^2'- n') cos (1/1'+ 1^2 )] (127) 



a 



A A 



— sin(i/2'-i^i')cos(i/,+ i/j') (1^2'- 1^1')+ (3/64) (i/j'-Fj') (4tan^ao- 1) 



2 4 



+ (1/16) (1-3 tan^oo) sin {v^- Vi) cos (1^1' + V2) 

 -(1/128) sin 2 (i/j'- i/,') [2 cos'(i^;+ 1^2') " H 

 Note that the first two terms of (127) are equivalent to Forsyth's equation, page 120 of 

 his treatise. 



Now for the value of c, we find on page 97 of Forsyth, that for approximations involving 

 f^ (second order in the flattening) a value of a that is accurate up to f inclusive must be 

 substituted in the first term of c. Hence from Hid, lllf, 111k we have 



c = 2f cos'a„ + 3f'cos'ao-4f'cos^ao(l+2A) . (128) 



This value of c placed in (127) with the value of A from (125) gives: 



85 



