NavList:

Thanks for responding.  As my answer is only 0.2' different than your in both latitude and longitude, I thought I knew why this was so, but I am apparently wrong.  This is what I thought:

The GHAs and Decs calculated by Celestial Tools for both sights appear to agree with the Nautical Almanac.  The differences lie in the Sun's main correction.  As I tell my students, we don't really believe that for a given limb and apparent altitude, the Sun's correction is constant for six months, changes abruptly, then is constant for another six months - this is just the way the Nautical Almanac packs a year's worth of data into a third of a page with no more than a couple of tenths of an arc-minute error.  In its default mode, Celestial Tools attempts to use a six-month average, but, after rounding, is often off by 0.1'.  It was off by 0.1' for both of these sights (1st sight:  NA -18.6', CT -18.7'; 2nd sight: NA +11.9', CT +11.8'), giving a 0.1' error in Ho.  I thought this might result in the differences in latitude and longitude.

But when I tried both Ho values for your answer and for mine, for both sights, using the Law of Cosines, I did not get what I expected. 

My L, Lo  47º29.0'N, 122º13.9'W

Your L, Lo 47º29.2'N, 122º14.1'W

1st sight

2nd sight

GHA 180º35.5'

Dec 10º40.3'S

My L, Lo with Ho   12º14.0' a = 0.0 nm

My L, Lo with Ho   12º14.1' a = 0.1 nm T

Your L, Lo with Ho 12º14.0' a = 0.0 nm

Your L, Lo with Ho 12º14.1' a = 0.1 nm T

Unless I screwed something up, for the first sight, my location looks better than yours regardless of the Ho used.  For the second sight, our results are the same regardless of the Ho used.  In any case, using the Almanac values of the Sun's main altitude correction, there would appear to be a third location solution:  47º28.9'N, 122º14.0' (or 13.9')W.  But this did not explain how you got your solution.

Then a light bulb went on.  I thought I knew what was going on.  The GHA values from the Nautical Almanac were "wrong".  You, and apparently the maker of the video, used a computerized or on-line almanac, which gives what I call an "accurate" value.  The Almanac, however, deliberately adjusts the hourly values of the Sun's GHA "by up to 0.15' to reduce the error due to ignoring the v-correction."  Celestial Tools has provisions for using both the "accurate" and Nautical Almanac values for the GHA of the Sun (the Nautical Almanac value agreeing with the printed Almanac in about 99% of the cases), but, for the dates and times of these sights, the values were the same.  Just to be sure, I checked a few computerized and on-line almanacs, including the USNO site, and found them all to be in agreement.  So that wasn't it.

So where did I go wrong?  How did you get your answer?  Does your Pilot Navigator disagree with any of the pertinent values?

FWIW, not really relevant to this problem, I also noticed that in the video the altitude correction table shown is from 2003.  The tables changed in 2004, so the corrections for 2003 are not the same as for 2010.  However, for the apparent altitudes of these sights, the corrections are the same for both years.

Stan