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So far I haven't seen any justification for the "most probable" position, much 
less for the use of an ellipse (which has 2 focii) rather than a circle (with 
one focii, i.e. center) when using 3 or more LOPs (which would indicate a 
need for 3 or more "focii").

Forgive me if the plural of focus is not focii, my latin isn't rusty, it simply "isn't".

A valid "most probable position" would have to take into account the 
individual variations and probable errors inherent in EACH of the LOPs. 
Graphically this could mean using a thicker pencil to draw each, i.e. if the 
accuracy of the sextant is 0.2 minutes and the accuracy of the eye on the 
horizon sight is another .2 minutes, draw the line to reflect the possible 
range of positions that are derived from being off .4 minutes to each side of 
the reading.

Once each line is drawn, the "most probable" position them becomes that area 
included under all three zones (the lines plus the extra width around them) 
and if all three zone do not contain a common area, it would still be most 
probable that the error in each site be expanded until the zones overlapped. 
At that point a circle (yes circle) or error could be expanded out to contain 
the probable positions, or an irregular complex shape that would be possibly 
more accurate than the circle.

I'd be able to do this graphically, but not statistically. I suspect the math 
would rapidly get more complex than the "use a thicker pencil and make a 
circle of error" approach, without giving much more useful information.

   

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