We now check equations (9) and (17), using again values for the Clarke 1866 spheroid and 



for the maximum value of A<^. 



From (9) and (12) v/e have 



sin ^(|> = 3.390074081 x 10'' sin 2(f) - 2.878029 x 10 '" sin 40 + 3.665 x 10'' sin 60. (18) 



From (12) and (17) find 



A0 (seconds) = 699':2540 sin 20 - 0':5936 sin 40 + TO004 sin 60. (19) 



Now with = 45° 02 55'1106 from (6), find sin 20 = + 0.99999856, sin 40 = - 0.00339575, 



(20) 

 sin 60 = - 0.99998703. 



The values from (20) placed in (18) give 



sin A0 = 0.0033900753 which checks the value found before in the 10th place. (See (6)). 



The values from (20) placed in (19) give A0 (seconds) = 6991' 2530 + :'0020 - :'0004 = 

 6991' 2546, or 11' 391' 255 which is the value of IS.4> ^^^. (See (6)). 



For explicit computation of as a function of d, we obtain the following development. From 

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



h + N = a cos 6/ cos = Ne^ + a sin 0/sin 0, whence 



tan = tan 9 + (e^/a cos 6) (N sin 0) 



(21) 



or tan = tan + (eV 1 + tan^ (^) (tan 0/\/ 1 + (1 - e^) tan^ ). 

 (NOTE: Equation (21) also follows directly from (3) by expanding the left hand side and 

 dividing every term by the product cos cos d. sin A0 = sin cos 6 - cos sin 6.) 

 Now (21) is of the form 



y = X + h (x) g (y) 

 and the Lagrange expansion formula may be used, [3] . 



Equation (21) may be written 



y = X + e^ (1 + x^) '/^ . y[ 1 + (1 - e^) yT'^' (22) 



Where y = tan 0, x = tan d, h (x) = e'(l + x')'/' , g(y) = y[l + (1 - e')y']"'/'. 

 By use of the Lagrange expansion formula, a function f(y) which has a power series 

 representation may be written 



ih(x)l" d 



n-1 



f(y) = f(x) + 2 — :i— ^ f'(x) 1 g(x)!" (23) 



n=l n ! dx n ~ J^ 



With y = tan 0, f(y) = arc tan y = ; x = tan 6, f(x) = arc tan x= 6, f tx) = = cos^^, 



1 + x^ 



equation (23) may be written 



A0 = - = £ ^'"^^^"^ 111 G(d) ■ (24) 



n=l n! d,(.n-l 



Where G{d) = (cos^ d) (tan ^/^Jl + (l- e^) tan^ (9) ", 6* = arc tan x. 



16 



