To find (1) the o'clock when the sun passes north, at Greenhithe. And
(2) the direction of the sun from Greenhithe at 10:06 AM (on 8 August).
Longitude of Greenhithe:
174 deg 42 min = 174.7 degrees East longitude
Sun above zero longitude at midday GMT, midnight NZST.
Sun above 174.7 degrees East longitude at 174.7/180 of 12 hours beforehand.
= 11.647 or 11 hours 39 minutes before midnight:
--== 21 minutes past 12-noon, the sun is N of us. ==--
(2) At equinox, the sun rises due E at 21 minutes past 6 AM,
and sets due W at 21 minutes past 6 PM, passing through N
at 21 minutes past 12 midday.
Call those 6.33, and 12.33 (the gnomon is not symmetrical).
So at 10.10 AM, the sun is (10.10-6.33 = 3.77 of its 6.00 hours, or
0.6283 of the 90 degrees between E and N.
So at 10.10 AM, the sun is 56.5 degrees less than 90;
it is 33.5 degrees E of N. "North East by North", on a compass rose.
{The next is not relevant to the above, only to the sun's elevation
angle.
On 8 August, 48 days after midwinter day (of 21 June).
There are 365.25/4 = 91.3 days from midwinter day to equinox.
The sun goes N and S over the year in SHM (ignoring asymmetry), and
taking the zero as midwinter day, its elevation angle changes as
a cosine.
So the angle change (from 23.5 degrees N of the equator on midwinter
day) = inverse cosine (48 day/91.3 day) = inv.cos(0.5257) = 58.3 degrees;
58.3/90 * 23.5 = 15.2 degrees up from -23.5 = -8.3 degrees.
With our latitude = 36.78 degrees (i.e. sun's rays come in
at ~37 degrees to the vertical when the sun is over the equator,
at equinox), on 8 August they were at 36.78 + 8.3 = 45 degrees;
at midday.}