NavList:

Robin Stuart, you wrote:
"I was recently motivated to look into this matter and found that in the log of Shackleton’s vessel, Endurance, an effort is clearly being made to take the time sights on the prime vertical whenever possible. I suspect that this is a required procedure in some Royal Navy manual that Frank Worsley was following."

That's interesting! And it makes excellent sense in those unique circumstances. Latitude was uncertain. Days are very long so the Sun will hit the prime vertical if it's out at all. But "high prestige" navigation, like the navigation during Shackleton's expedition is its own universe. Your suspicion that this was a recommended Royal Navy procedure is worth investigating, but it's only one possibility. 

For what it's worth, I have examined something like 500-1000 individual time sights (I don't keep count) from the 19th century through the early 20th century, all from US and some French commercial vessels, and in that large (culturally limited!) sample, there is a standard practice of taking sights at some fixed time of day, typically 9am and/or 3pm. I should emphasize again that navigation is a cultural activity. There are "schools" of navigation. It may well be that at the end of the 19th century, as RN navigators were taking up the "modern" intercept method, they were also being advised that former traditional practice was "wrong". Playing up the supposed flaws of the "Old Navigation" may have been a useful means of weaning navigators from those reliable, time-honored methods.

We also have to watch out for "backfilled" hypotheses which try to airbrush history to fit a theoretical model. The math provides such nice evidence that taking sights on the prime vertical is best for longitude, so that's what they must have done. Unfortunately, theory is not history. Navigators did not normally worry about time sights taken "dead on" the prime vertical in the cases that I have examined, and (turning to theory!) such sights are obviously impossible for half of the year in most parts of the world in any case. It was by no means part of the standard approach to celestial navigation. It certainly wasn't "the norm" which you might expect from many descriptions of the process in recent decades. 

Your hypothesis about Royal Navy practice is worth investigating. You/we/NavList (?) might try to examine some logbooks and other papers from RN vessels c. 1900. Original calculations are frequently found on bits of scrap paper and on endpapers in old books. The circumstances of the sights can often be deduced from the scribblings. The key to history is primary source evidence.

Of course, it should also be said that it's a trivial matter, a small bit of detail. Time sights for longitude, regardless of cultural practice, were not and are not dependent on the prime vertical except in unusual circumstances.

Frank Reed