2 4 o C 



= — sin 2cb H sin ^c6 cos <A h e* sin ^<i> cos d) + — e' sin'oS cos <h . (7") 



2 28 16 -r ^ Ml 



now sin'0 cos ^ = J^ sin 20 - '/^ sin 40 



sin '0 cos (j) = ^/^2 sin 20 - '/^ sin 40 + '/^2 ^i" 6 (8) 



sin '0 cos = '' /(^i sin 20 - '^4 sin 40 + ^^4 sin 60 - V128 sin 80, 

 and the values from (8) placed in (7) give 



sin A0 = c^sin 20— Cj sin 40 + c 3 sin 60 — C4 sin 80 ; 

 2 4 



where c, = |- + |-+ ''/,,, e' + "/,024 e\ c, = eV,e + '/,, e' + "/,o,, e» , (9) 



^3 '25 6 ® '1024 S » '- 4 /2O48 ^ 



If A0 in radians is desired rather than sin A0, then in the expansion 



arc sin x = x(l + x^/^ + ) (10) 



let X = sin A0, whence arc sin x = A0 and 



A0 = sin A0 (1 + ^H^ + ). (11) 



6 



from (9) with e' = 0.006768657997, find 



c, = 0.003390074081, c^ = 0.000002878029, 



c, = 3.665 X 10"', C4 = 5 X lO"'' (negligible). 

 For estimation purposes the values in (12) may be written 



Ci = 3 X 10'% Cj = 3 X 10"% C3 = 4 X 10"' 



c,' = 9 X 10'% cj' = 9 X 10-'% C3' = 2 X 10-''. 



With the value of sin A0 from (9) in terms of the estimation coefficients (13) we examine 



the term (sin'A0)/6 in (11), and find that (11) may be written A0 = sin A0 + 

 3 2 



-^ sin ' 20 — - sin '20 sin 40. (14) 



6 2 



since sin^20 = % sin 20 - ^ sin 60 



sin'20 sin 40 = V? sin 40 - '4 sin 80, (15) 



equation (14) may be v/ritten, with the value of sin A0 from (9), as 



/ ci'\ / c.'c2\ / cA 



) (radians) =((ci + ))sin 20 - (c^ + — — ^Isin 40 + | C3 -""^l sin 60, (16) 



A0 (seconds) = (206,264.8062) A0 (radians), 

 where Cj, C2, Cj, are given by the expressions in (9) in terms of the eccentricity of the meridian 

 ellipse. 



(12) 



(13) 



15 



