Note that the term C4 sin 86 does not contribute to the result. Also, only eight place tables 

 of trigonometric natural functions were used, [4] . 



Hence for geodetic latitude (p corresponding to geocentric latitude 6 on the auxiliary sphere, 

 the following formulas are sufficient for any spheroid of reference to 0.001 second: 



A(f> (seconds) = </> - 6 = (206,264.8062) (C, sin 26 + Cj sin id + C3 sin 66) 



C, = 'Ae' + (l/8)e''+(ll/256)e'+(31/1024)e% C^ = (3/16)e'' +(5/64)e' +(25/1024)e% (37) 



C3 -(77/768)e'+(59/1024)e',e is eccentricity of the meridian. 



Now we have noted that the geocentric latitude 6 as defined here is called the parametric or 

 reduced latitude in geodetic nomenclature and has a corresponding geodetic latitude <p„ as shown 

 in Figure 1. From (l) we see that they are related by the equation tan (/>„ = (tan 6)/\/l — e^. (38) 

 For instance from (6) for 6 = 44° 51' ISI'SSl find from (38) that 0o = 44° 57' 06:'069. Also from 

 (6), = 45° 02' 55."106, whence for 6 = 44° 51' 15:'851 we have A(^„ = - ^„ = 0° 05' 49r037.^^^^ 



Using the values from (34), equation (37) may be written for the Clarke 1866 spheroid as 

 A0 (seconds) = - (9 = 6991'2520 sin 26 + i:'7769 sin 461 + 0r0064 sin 66. (40) 



From C. & G.S. special publication No. 67, [5], find 



0„ - = 350':2202 sin 26 + 0':2973 sin 46 + OIOOOS sin 66. (41) 



Subtracting (41) from (40) one finds 



A0O = (^ - <;6„ = 349': 0318 sin 26 + V'A796 sin 46 + 0':0061 sin 66. (42) 



With 6 = 44° 51' 15':851 and the values from (28), equation (42) gives 



A<;6o = 5' 49';036 which is within 0.001 second of (39) . 



From the second and third members of each set of equations (2) find 



h = a sin esc .jS - (1 - e^) N = a cos ^ sec - N. (43) 



To develop h in a power series in (p, free of N and 6, refer again to Figure 1. If the 

 tangent at meets OP in P', then PP'= a - (a^/N) sec Acf), h = PP'cos A0, whence 



h/a = cos Acf) - a/N = cos Ac^ - \J I - e' sin^c^ (44) 



With cos A(^ = yj 1 ~ sin^A(j!), and the value of sin A(p from (3), (44) may be written 



h/a = (1 - e' sin= <;6)-'/' {[1 - e' sin' <f> {I + e' cos' <?!,)] '/'- 1 + e' sin' cf> \ . (45) 



The relation (45) may also be obtained directly from equation (2) by eliminating 6 



between the equations a cos = (h + N) cos <f) and a sin 6 = [h. + N(l — e') ] sin (p. 

 Expanding the two radicals by the binomial formula, (45) may be written 

 h/a = (e72 - eV2) sin'0 + [(5/8)6" - 'Ae' - (l/8)e'] sin V 



+ [(9/16)e' - (l/4)e'] sin V + (53/128)e' sin V 



(46) 



19 



