Making the identity substitutions 



sinM = V2 - V2 cos 2d, sinM = (3/8) -y2Cos2d+ (cos 4d)/8 



sin'd = (5/16) - (15/32) cos 2d + (3/16) cos4d - (1/32) cos6d, in (39) and integrating 

 term by term according to (38) one obtains 



s/a = (d. + d,) - 'AV^'VA (di + d^) - ^(sin 2d, + sin 2d,)] - (1/8)1^" [(3/8) (d, + d,) - 



!4(sin2di + sin 2d2)+ (1/32) (sin4d, + sin4dj)] -(l/16)k' [(5/16) (d, +d2) - (40) 



(15/64) (sin2d, + sin2d2) + (3/64) (sin4d, + sin4d2) - (1/192) (sin6d, + sin6dj)]. 

 By means of the identity sin x + sin y = 

 2 sin /4(x + y) cos/4(x — y), equation (40) may be written finally as 

 s/a = (d, + d,) - %V^ {{A, + d,) - sin (d. + d,) cos (d, - d,)] 



-(l/128)k'' [6(di + dj) - 8 sin (d, + d,) cos d, - dj) + sin 2(di + dj) cos2(d, -d,)] (41) 



- (l/1536)k' [30(di +dj) -45 sin(di+dj) cos (d,-dj)+9 sin2(d,+dj) cos2(di-d2) 



- sin 3(di + d,) cos 3(d, - dj)], 



a and e are semimajor axis and eccentricity of the meridian ellipse, k = (e\,'l — e^/a) N,, sin^p 



(k = e^, the eccentricity of the great elliptic arc), ^0 is the vertex of the great elliptic arc as 



given by (21). d, = arc cos (N, sin (;6,/No sin (f)g), d, = arc cos (Nj sin (^Sj/Nq sin (pg). When 



(f>o ~ 90°; equation (41) gives a meridian arc of the spheroid. When <;6o = 0, an arc of the 



equator or circle of radius a is given. Formula (41) thus consists of a circular arc and successive 



corrective terms. 



To examine the contribution of the terms in (41) take the case cfji = (/>2 = 0, 0o= 45°, 

 dj = d, = 90° which will give the semilength of the great ellipse making an angle of 45° with 

 the equator. For the Clarke 1866 spheroid, e' = 6.768657997 x 10"', a = 6,378,206.4 meters. 

 From (41) we have then 



1st term ax(d, + d2)= 20,037,773 meters 



2nd term -a x 2.65804 x 10" = - 16,954 meters 



3rd term -a x 0.17 x 10'= = - 11 meters 



4th term -a x 0.24 x 10'^ = - 0.015 meters 

 When (f)a - 90, 4> 1= (f>2 == 0, dj + dj = tt, and (41) reduces to the usual formula for length of 

 the semimeridian from equator to equator through the pole s = a;7[l — /4e^ -(3/64)e^ -(5/256)e J. 



52 



