DIFFERENCE FORMULAS TO SECOND ORDER IN THE FLATTENING 

 d'- d = - (f/2) Y sin d + (f yi6) [4Y (X - 3) sin d + (2Y^ - X^ ) sin 2d], 



= -(f/2) Y'sin d'-(fVl6) [4Y'(X'-l)sind'+(2Y'^-X'^)sin2d1 ; 



X'- X = fX (2 - X) il + (f/2) (3-2 X)K 

 = fX'(2-X')!l-(f/2) (1 -2X')h 



Y'-Y =fY(2-X) + (f/2)(X'-Y^)cos d 



(P/8 



4Y (2 - X) (3 - 2X) 



+ (X^-Y^) i(ll -5X) cos d+Y (1 -3 cosM)! 



= fY'(2 - X') + (f/2) (X'^- Y'^) cos d' 



(fVS) 



4Y'(2-X') (1 -2X') 

 + (X''-Y'') i 2(5-3X0 cos d'+Y'(l-3 cos'd')^ 



1/2 



CHORD DISTANCE, c 



c = a Hi -cos (d, + d2)S !2-k'[l -cos (d. - d J] sl 

 Where in terms of geodetic latitude 0, 



dj = arc cos (N^ sin (ji^ /N,, sin <;6o), dj = arc cos (Nj sin cfi2 /Ng sin cj)^) 



k^ = [eMl-e^)/aMNo^sinV„ 

 in terms of parametric latitude 6 



d 1= arc cos (sin di /sin 6^), dj = arc cos (sin 02 /sin 6 a), k^ = e^ sin ^ d 



ANGLE OF DIP OF THE CHORD, /3 



( (l-e^)[l-cos(d, + dj] 1 '/^ 



I [2 - kM 1 - cos (d^ - d, ) !] (1 - e^ + k^ cos^ d.)j 

 with k, di, dj expressible in terms of either geodetic or parametric latitude as given above. 



MAXIMUM SEPARATION OF CHORD AND ELLIPTIC ARC, H„ 



Ho =1^ sin '/2(d, +d,)[l-COS ^2(d, +d,)]. 



where c is the chord length as given above, bo = ay 1 — k^ ; c, k, dj, d2 expressible in either 

 parametric or geodetic latitude as given above. 



GEOGRAPHIC COORDINATES OF POINT OF MAXIMUM SEPARATION 



tan (f> = R/D, or cos 2<^ = (D' - R')/(D' + R' ), tan A = (cos d^ sin AA)/(cos 6, + cos d^ cos AA), 

 R = sin 01 + sin 6*2 , D = (0.996609925) (4 cos' K2d-R')vf d is spherical distance between the 



points P, (^1 , Aj), Pj (02, Aj) on the ellipsoid, d is parametric latitude, AA = A2— A,. See Figure 23 



for sample computation. 



11 



