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Here is my attempt at the problem.

Wellington   41° 16.706'  S     174°  46.6' E

Auckland  36 °  50.9' S    174°  45.7' E

dlat  4° 25.7'  =  265.7'

dlong    0.9'

Mean lat  38°   38.8'

From Norie mean lat to middle lat   -14'

Hence middle lat = 38°   24.8'

dep = 0.9 x cos (38° 24.8')   = 0.7'

Bearing =  arc tan ( 265.7/0.7)   = 89.9°

Distance -   265.7/sin (89°.9,)  = 265.7  to 3 decimal places.

Next I performed a rough sanity check with google-earth which gave   distance   271 nm,  bearing 0°

The dip from Wellington to Auckland  is therefore   4°  25.7' /2   =   2°   12.9'

From google I found that distance - 2 R sin (subtended angle). Assune a  perfect sphere  R = 3444 nm.

d = 2 x 3444 x sin (2° 12.9')  =   266nm

But this is the length of the rumb line!!!!!!!!   There are two possible explanations - the small subtended angle requires that I work to a higher resolution or I have made a big blunder.