NavList:

A couple of quick answers:

• The logarithmic calculation portion of Thompson's method only covers the linear component of the Moon's parallax in the clearing process. This is the lion's share. Everything else is small and gets dumped into the table, which is what makes this method and its numerous cousins convenient.
• The tabulated portion or "third correction" thus includes the refraction for both bodies as well as two quadratic terms. Note that a modern computation could do this differently: calculate the exact clearing value by the direct triangle approach and then subtract the value obtained from the linear component of the Moon's parallax. The residual is the same as the third correction. Historically, starting with the bumbling Baron F. X. von Zach, this was what many nineteenth century commentators imagined that Thompson had done. He himself kept quiet.
• The blanks in the third table are impossible geometries, which we discussed in my "Lunars" class (and you probably remember). For example, suppose the Moon is 45° high and the Sun is also 45° high. The maximum lunar distance is then 90° which occurs when the two bodies are exactly opposite each other (opposite in azimuth). All other distances below 90 are possible, but 100° cannot occur for those given altitudes. The table is blank for that combination, and any greater distances.
• Note: there has been a lot of junk written about Thompson's tables, going back to the 1820s and including here in NavList messages in the 21st century. The tables were successful commercially. Our naturally geeky crowd is relatively blind to commercial success and, as folks with engineering backgrounds, some imagine mathematical details must underly any success. Jan Kalivoda who wrote extensively about Thompson's tables with much help from George Huxtable, and they unfortunately fell into this trap. Thompson's tables are kin to Martelli's "mystery" tables --part of their success came from misguided reviewers who imagined magical mathematical complexity when there was only hard work in compilation and publication.
• Finally, the real weakness in Thompson's tables is that the third correction has been calculated for standard temperature and pressure and for a mean value of the Moon's parallax. It's interesting to consider how much of a concern that might be. 

Frank Reed
ReedNavigation.com
Conanicut Island USA