NavList:

David, you wrote:
"If the lines were from different navigational aids, e.g. an NDB line, a VOR line, crossing a railway line, and a sun line, three of which had been transferred up along track, the bands of error would be different for each line."

I believe Greg's question was specifically about celestial lines of position. We know how to handle three correctly. Is there a corresponding semi-graphical trick for four? There is, of course, an analytical method for N lines (standard instructions have been included in the Nautical Almanac since the 1980s), and that certainly gets the job done, but it doesn't offer much insight.

Let's set aside for now the question of different sources of navigational information (it's an interesting question, and we should come back to it later). I encourage you to stick to one narrow topic in this particular broad subject since, as we have seen in the past, it can quickly degenerate into different people talking about somewhat different things and getting quite frustrated with each other. 

You also wrote:
"Of course the true position doesn’t even have to lie inside the cocked hat."

Here we run into an enormous source of confusion. Many navigators have heard that there is a greater probability that the actual position is outside the "cocked hat" and so then they start trying to figure out "where" it really is outside. Then someone will bring up a case of systematic error, and "aha!" they all say, "there really is a way to figure out where it is outside the cocked hat!" But this is all just confusion. The most likely location for the actual position is within the cocked hat and the error ellipse has some size and shape around that position. And the key is to understand that the error ellipse is not determined naively by the shape of the cocked hat. I guarantee you that most people who studied a little celestial navigation and worked with plots of lines of position will get this wrong. Utterly, completely wrong!

David, you wrote:
"It’s fairly easy to show that for three relative bearings on the same side of the aircraft say 030, 090, and 150 for an aircraft with 5degrees compass error, say compass heading = 005 actual heading 360, the aircraft’s true position lies outside the cocked hat."

Do you happen to remember where you learned about this particular example? It is a source of great confusion that sadly seems to have become important lore among many navigators, and in my experience, especially so among former military navigators. It is true that a known systematic error can be eliminated by collapsing the "cocked hat" and the result will frequently yield a revised position outside the original "cocked hat" or triangle. This is true. It is absolutely, definitely true. So if (and this is a huge "if")... if a navigator knows that there is systematic error in three LOPs, then it is possible to eliminate it and improve upon the original estimated position within, or outside!, the triangle. Unfortunately, this specific, specialized case was grossly mis-applied by many navigators as a means of improving any three-body, or three LOP, fix. This trainwreck has happened more than once in NavList discussions.

David, you wrote:
"Therefore, the moral is take position lines, not just with a good cut, but also from both sides of the aircraft."

That is excellent advice in any case, but is it relevant to the original question that Greg posed in this thread? There is a world of difference between systematic and random error. A technique designed to reduce systematic error should be applied only if there is a positive reason to believe that systematic error exists or if we have the luxury of a reasonably large sample of sights (see below).

You concluded:
"I’ll let you think about it for a day and provide a drawn answer tomorrow evening if no one’s beaten me to it."

It's actually a well-known story, David...  

There is also a more interesting case from the perspective of modern (perhaps automated) navigation. If you have multiple LOPs, how do you handle the possibility of systematic error combined with random error? Robin Stuart has already alluded to this in his message. It is possible to solve mathematically for any systematic error if we have enough data. For an example that's not hard to understand even for beginning navigation students, consider a navigator shooting three stars, separated by the optimal 120°,  who has no data on index correction. If we shoot one sight of each body, we get an equilateral triangle of crossing LOPs. The fix should be placed at the center of the triangle, and that's that. The size of the triangle provides no useful information (except in a theoretical case where these are the very first sights ever taken and in that case the size yields the first estimate of random error in the sights), and no procedure should be applied to reduce the triangle's size. However, if the navigator is quite sure that the missing index correction is substantial (in other words, we have a strong reason to believe that there is a substantial systematic error), the spacing of the LOPs might be reduced by including some estimate. We can "collapse the triangle". It is a mistake, however, to do this with three sights unless we know that there is a systematic error. But suppose we shoot those same three stars multiple times in rapid succession, maybe a dozen sights of each star. This could be realized in practice with an automated system or with a small team of student navigators working together (a dozen properly-trained navigators shooting those three stars in succession). Now when we plot the sights, we will find one of two rather distinct patterns. Either the three dozen LOPs all fall more or less on top of each yielding a tight "tangled knot" of LOPs, or we discover that the LOPs from each individual star fall together in a bunch but that the collection of LOPs from all three stars leaves a sort of "donut hole". They form separate bundles. In this case, it's not hard to see that we can transform the "donut hole" case to the "tangled knot" case by applying a constant offset, in other words a systematic error correction, to all of the sights "collapsing the donut hole". This is a case of the same procedure that Robin Stuart mentioned. When there is enough data, the systematic error can be treated as another unknown.

Frank Reed
Conanicut Island  USA