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    Re: determination of longitude and the prime vertical
    From: Bill B
    Date: 2007 Apr 11, 21:13 -0400

    > From: Geoffrey Kolbe 
    >
    > I would question Bill's statement that "the skew of the LOP helps to
    > null out error".
    
    And rightly so. I was trying to be in two places at once, and made several
    conceptual errors. In this case plotting at N41 would cause me to be further
    off true position by a hair.  If the intercept was plotted away and the Az
    was less than 270d, the relationship to true position *may* improve.  It is
    however a moot point. Why would I plot an intersection at the LOP and N41
    and use a N40 Hc if I now believed myself to be at N41?
    
    Fuzzy thinking diagram attached.  Only the intercept is truly significant.
    
    Using the scenario where I believed I was at N40 W60 when I was actually at
    N41 W60 (the examples I gave in a prior post),  I calculated the moment of
    AZ 270 and altitude for N40 W60. Hc would be 24d 0!2.  Because I was
    actually at N41 W60, my corrected observed altitude, Ho (in a perfect
    world), would be 24d 0!0 at the time calculated for Az 270 at N40 W60. My
    intercept would have been 0.2 away from the body.
    
    > Where the sun is near the Prime Vertical (within a degree or so), It
    > would seem that the best scenario is to assume the azimuth is exactly
    > 270 and draw a position line that is exactly North-South. Such a
    > method will be more tolerant of large errors in latitude for the
    > estimated position than the classic position line method.
    
    Agreed in practice. In my scenario the Az from using Hc 24d and N40 W60 AP
    was 270d 0!2 instead of predicted 270.00.  Try to plot that LOP precisely!
    >
    > "And when the Latitude is uncertain, there are only two proper
    > instants per day (when the sun bears exactly East or exactly West)
    > when the solar altitude can be used to find the Longitude by
    > Chronometer with accuracy.  If one makes an observation other than at
    > those times....... the unavoidable errors in measurement may
    > cause frequent errors which are very great."
    >
    > I have to ask whether Sumner was right...?
    
    First put numbers to "accuracy," "very great," and "Latitude is uncertain."
    As we have seen in my mid latitude examples the error from being off true
    latitude by 60 miles and 00:02:42 from Az 270d time is an intercept of 0.2;
    equal to or better than the accuracy of the system in practice.
    
    Sumner is also adding "unavoidable" measurement error on top of any
    theoretical argument.  Again not quantified.
    
    In any case Sumner was using a time sight for longitude or constructing an
    LOP. In my trials I have found the time-sight method is more sensitive to
    being off Az 90/270d than the intercept method.  Nominally any error from
    not being spot on 270 and AP latitude was a bit over 50% greater for the
    time sight when compared to the error from the intercept method (not
    available to Sumner).
    
    Another factor, albeit relative, is as the latitude increases the conversion
    from nautical miles or arc minutes latitude to arc minutes longitude becomes
    more of a factor.  0.2 miles or minutes at latitude 0 = 2!0. 0.2 miles at
    N40 = 2!6 and at N70 = 5!8. (Roughly, see Bowditch for precise conversion
    values.) Near the poles a 0.2 error could yield an alarmingly high value for
    longitude.  In the bigger picture, it is still approx. 0.2 miles off.
    
    Thank you for prompting me to rethink.
    
    Bill
    
    
    
    
    
    
    
    
    
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