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I have a question about the tables used to correct ex-meridian sights:

In bowditch (1995) tables 24 and 25 are used for ex-meridian altitudes
(previous editions used tables 29 and 30 I believe). Table 24 gives the
amount by which the altitude of a body changes per minute of time either
side of meridian transit. This change (called the  factor 'a' and is in
seconds of altitude) is then entered into table 25 to give the actual
corection for the altitude.
This next table is based on the formula:  Correction = a X t X t/60  (in
words: 'a' multiplied by 't' squared then divided by 60). Where 'a' is
extracted from table 24 and 't' is the difference in time between the
observation and meridian passage
The divide by 60 is just for the seconds/minute relationship since 'a' is in
seconds and 't' is in minutes.
My question is: Why is t squared ?? why is the formula not simply:
Correction = a X t/60 ?? (since 'a' is said to be the change per minute; see
below...)
regards
Russell

Extract from Bowditch describing table 24, from which the 'a' factor is
extracted...

Table 24. Altitude Factors - In one minute of time
from meridian transit the altitude of a celestial body changes
by the amount shown in this table if the altitude is
between 6� and 86�, the latitude is not more than 60�, and
the declination is not more than 63�. The values taken from
this table are used to enter table 25 for solving reduction to
the meridian (ex-meridian) problems.
The table was computed using the formula: a = 1.9635" cos L cos d csc( L ~ d
)

   

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