And when using the sun line landfall approach, any accumulated uncertainty in the D.R. collapses into a line 14 NM thick (7 NM uncertainty on each side of the plotted LOP) with the line being as long as twice the reasonable maximum uncertainty in the DR at that point.

I understand what you are saying but I do not understand how the accumulated uncertainty collapses. For example, lets say they were 400NM out when they obtained their last fix, how does 40NM collapse to 14NM along the line? I would suspect a total possible DR error with a radius of 54NM and I do not see how this is reduced to a 14NM strip.

A picture is worth a thousand words.

Diagram A shows the DR uncertainty after flying 400 NM starting from a pinpoint visual fix with basically zero uncertainty in the starting fix. The uncertainty increases one mile for every ten miles flown so after 400 NM you can draw a circle of radius 40 NM around the DR position and you can be very certain that you must be inside that circle and you are much more likely to be near the center than near the edge of the circle. (In a very rare case you might be outside of the circle but, even then, you will be very near to the circle.) Since you know that you are inside that circle you have excluded the entire rest of the world as your possible location.

Diagram B shows the same thing but starting from a celestial fix that has the standard 10 NM uncertainty. Since you might actually be near the edge of that circle in any direction, this 10 NM uncertainty must be carried forward in figuring the uncertainty of a DR position so you add the original 10 NM uncertainty to the 40 NM uncertainty resulting from dead reckoning for 400 NM, so you draw the circle of uncertainty with a 50 NM radius.

Diagram C shows a celestial LOP with the band of uncertainty on each side. These bands of uncertainty are just like the circles of uncertainty around the DR in that they exclude the entire world outside the uncertainty bands. You know that you must be somewhere within the bands, you can't be anywhere else on earth.

Diagram D shows a celestial fix consisting of two LOPs that cross, along with their respective uncertainty bands. Since you must be within the uncertainty bands along LOP "A" and, at the same time, you must be within the uncertainty bands along LOP "B," the only place you can be on earth is where these two areas overlap, the rest of the world has been eliminated. Since the uncertainty of celestial fixes is taken as 10 NM, and this represents the corner of the overlap, the uncertainty of the two LOPs cannot exceed 7 NM, (ask Pythagoras why this is true.) (The diagram shows the ideal case with the two LOPs crossing at right angles which results in a circle of uncertainty around the intersection of the LOPs. When the LOPs cross at a different angle (known as the "cut") then the area of uncertainty is actually an ellipse but for normal cuts you can just consider that they are also circles.)

Diagram E shows the uncertainty circle surrounding a DR position which eliminates everything outside the circle. Imposed on top of this circle is a celestial LOP. Since you can only be within the area of overlap between the LOP uncertainty band and the uncertainty circle around the DR position, this then eliminates everything outside the LOP uncertainty bands and thus eliminates the two half-moon slices of the circular DR uncertainty area. So with this LOP, and the knowledge of the length of the DR leg, you know that you are within the 14 NM wide LOP uncertainty band and that the length of that LOP is limited by the diameter of the DR uncertainty circle.

So that is why the DR uncertainty circle collapses into a band 14 NM thick and 100 NM long (in this illustration.)

gl