X2 = N2 COS (^2 '^OS A^ 



72 = Nj COS 02 sin A A 

 z, = N, (1 - e^) sin di. 



and the y-axis is then in the plane of the equator — the xy-plane is the equatorial plane of the 

 ellipsoid. In this coordinate system the points Qj (<^i,Ai), Q^{(f)^,X^ have the rectangular 

 coordinates: 



Qji Xi = Nj cos cf>i Q2 



y, = 



Zj = Nj (1 - e^) sin <^, 

 The rectangular equation to the ellipsoid is 

 (1-e^) (x^ + y^) +z^-a^(l-e^) = 0, 

 where a, e are respectively the semimajor axis and eccentricity of the meridian ellipse. 

 The tangent plane to (2) at any point (xj, y,, zj is 

 (1 - e') (xxi + yy,) + zz. - aMl - e') = 0. 

 Hence the tangent plane at Qj is, from (1) and (3) 

 xN, cos 961 + z N, sin (f)^ - a^ = 0. 

 The equation of the plane containing the normal at Qi and the point Q2 is determined by 

 Q2 and the points (NjC^ cos 0i, 0,0), (0,0, - N,e^ sin^i), see Figure 13. With the coordinates 

 of Q2 from (1) we can write the equation as 



X y z 1 



N2Cos</)2 cos AA N2C0S <^2 sin AA N2(l-e^) sin0j 1 



Nie'cosv^i 1 



-NiC^sincjii 1 



which upon expansion may be written 



Ax + By - Cz - D = 

 where A = N2 sin 96, cos (^2^1^ AA 



B = (Ni sin (f)^ - N2 sin (^2) e^cos<;6,+ N^ (sin c^j^os <;6i-sin 961 cos 02^03 AA) 

 C = N2 cos 01 cos (p2 sin AA 

 D = NjNje^ sin 0i cos 0i cos 02 sin AA. 

 Now the direction cosines p, q, r of the intersection of two planes A^x + B,y + CjZ = D,, 

 AjX + B2y + C2Z = D2 are given by 



p = (B,C2 - B2C,)/d, q = (C A - A.C2)/d, r = (A,B2 - A2B,)/d 

 where d = [{Bfi^ - B,C2)' + (C,A2 - A,C2)' + {Afi^ - A^B.Y]'^'. 



Note from figure 13 that the tangent, tj, to the meridian at Qj lies in the plane y = and 

 that defined by equation (4). To apply (6) to these two planes we have respectively 

 A, = C, = Di = 0, Bj = 1; Aj = Ni cos 0^, B2 = 0, C2 = N^ sin 0,, D2 = a' and (6) gives the 

 direction cosines of t^ as pi = sin 0^, q^ = 0, r2 = - cos 0i. 



(1) 

 (2) 



(3) 



(4) 



=0, 



(5) 



(6) 



(7) 



44 



