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    Re: David Thompson's Navigational Technique
    From: Ken Muldrew
    Date: 2004 Jun 1, 11:39 -0600

    On 31 May 2004 at 22:24, Frank Reed wrote:
    > In case anyone's interested in the "mathy" side of this, this is the
    > quadratic or "Q" correction that I talked about a few weeks back under
    > 'Easy Lunars'. Moore's table 10 lists the quantity (1/2)*x^2*cot(d)
    > where x is any angle and d is the lunar distance. The quadratic term
    > "Q" in the series expansion is (1/2)*(dh_moon)^2*(1-A^2)*cot(d) where
    > dh_moon is the Moon's altitude correction and A is the Moon's "corner
    > cosine" (the percentage that tells us how much of the Moon's altitude
    > correction acts along the arc of th measured lunar). By multiplying
    > this out, you get
    >     (1/2)*dh_moon^2*cot(d)  -  (1/2)*(dh_moon*A)^2*cot(d)
    > so by entering table 10 twice, first with the Moon's altitude
    > correction and then with the correction multiplied by A (which we've
    > already calculated!), you get the complete quadratic correction.
    Maskelyne's procedure for the quadratic correction (sans table) is:
    {begin quote}
    For computing a second, but smaller correction than the first,
    necessary to be applied to the observations of the distance of the
    Moon from a star on account of parallax.
    Call the principal effect of parallax, found by the preceding rule, the
    parallax in distance; and find the parallax answering to the Moon's
    altitude. Then to the constant logarithm 0.941 add the logarithm of
    the sum of the parallax in altitude and the parallax in distance, the
    logarithm of the difference of the same parallaxes, and the cotangent
    of the observed distance of the Moon from the star (corrected for
    refraction, and the principal effect of parallax), the sum, abating 13
    from the index, is the logarithm of the number of seconds required,
    being the second correction of parallax; and is always to be added
    to the distance of the Moon from the star, first corrected for refraction,
    and the principal effect of parallax found above, in order to obtain the
    true distance; unless the distance exceeds 90 degrees, in which case
    it is to be subtracted.
    {end quote}
    If I'm reading this right, he has
    8.73 * (dh_moon + A) * (dh_moon - A) * cot(d) * 1/2
    which reduces to
    8.73 * [(1/2)*dh_moon^2*cot(d)  -  (1/2)*A^2*cot(d)]
    I don't really get the 8.73 term and he's left out the dh_moon from
    the 2nd term in square brackets. I'm assuming this latter difference
    is for calculational ease but is the 8.73 term somehow meant to
    Ken Muldrew.

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