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JCA:

I computed LAN= 12h 9m 42s by the equation of time method. What am I doing
wrong?

On 2008 May 16 meridian passage is 11h 56m and ET 12h is 3m 38s.
Subtracting ET from GMP I get 11h 52m 22s.

 

Your error John, is that you are applying the equation of time twice.  The entry "Meridian passage" in the Almanac gives you the Local mean time (LMT) of meridian passage to the nearest minute.  This shows us whether to add or subtract the equation of time; in this case we will subtract.  The equation of time entry gives us the actual difference between LAT and LMT of meridian passage of the sun to the nearest second.  In this case 1200 Local apparent time (LAT) minus 3m 38s to give us 11h 56m 22s for the LMT of mer. passage.  This of course must be converted to zone time, which you have done in the next step correctly.  I must note that I am unfamiliar with the acronym "GMP" - JCA

For Long 145d 40m E (-9h 42.6m) I get LMP 2h 9m 42s GMT.

For Zone -10 this becomes 12h 9m 42s.

My other results are

Az=0
Ho= 86d 5.8m
Hc= 86d 4.2m
1.6 toward
Lat= Dec-(90-Ho)= (19d 9.6m) - (3d 54.2m) = 15d 15.4m N