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594 PHIPPS’s JOURNAL:
«¢ I have calculated the longitude from each fet of obfervations feparately, to fhew how
near they agree with each other, and what degree of precifion one may expect in fimi-
lar cafes.
<< Obfervations of the diftances of the moon and fun, or ftars, may be ufeful to inforny
us if the time-keepers have fuffered any confiderable change in their rate of going. For
if the longitude deduced from the moon differs above two degrees from that found by
the watches, it is reafonable to imagine that this difference is owing to fome fault in the
watch, as the longitude found by lunar obfervations can hardly vary this quantity from.
the truth; but if the difference is much lefs, as about half a degree, it is more probable
that the watch is right, fince a fmall error in the diftance will produce this difference.
“¢ The diftances of the moon from Jupiter were obferved, becaufe Jupiter is a ver
bright object ; and the obfervations are eafier and lefs fallacious, particularly that of the
altitude, than thofe of a fixed ftar, whofe light is much fainter. This method, however,
requires a different form of calculation, from that of the obferved diftance of the moon
from a fixed ftar, whofe diftances are computed for every three hours, in the nautical
Almanac. The principal difficulty in the calculation is to find the moon’s longitude
from the obfervation of the diftance. This I have endeavoured to facilitate by the fol-
lowing problem, which may be applied to any zodiacal f{tar, and will be of ufe when the
ftar fet down in the ephemeris cannot be obferved.
« PropLem.—Having given the diftance of two objects near the ecliptic, with their
latitudes, to find their difference of longitude.
“ SonuTion.—Find an arc A, whofe logarithmic fine is the fum of the logarithms
of the fines of the two latitudes and the logarithmic tangent of half the diftance, rejeét-
ing twenty from the index of the fum.
“ Find an arc B, whofe logarithmic fine is the fum of the logarithmic verfed fine of
the difference of latitude, and the logarithmic cotangent of the diftance, rejecting ten
from the index of the fum.
“‘ Then A added to the obferved diftance, and B fubtra<ted from the fum, leaves the
difference of longitude.
<< If one of the latitudes is fouth, and the other north, the fum of the two arcs A and
B fubtracted from the diftance, leaves the difference of longitude.
«“ Exampie.—Augutt the thirty-firft, the obferved diltance of the moon’s center
from Jupiter, cleared of refration and parallax, was 32° 35/ 52’, the moon’s latitude
being 1° 47’ N., and that of Jupiter 1° 36’S.
‘« Latitude D 1° 47/ Sine 8,4930 Difference of latitude, 3° 23’ Vers. Sin. 7,2413
Lat. % - 1 36 Sine 8,4459 : ‘
Half diftance 16° 18 Tang. 9/4660, Diftance 32 36 ©-Cotang. 10,1941
Are A.o! 52” - Sine 26,4049 Are B! 9! 25” - - - ~ Sine 1
The fum of thefe ares. —10’ 17” Subtracted from 1 Meee
the diftance - - 432° 35 52
—-
leaves 32 25 35 the difference of longitude between the moon and Jupiter.
“« Knowing the longitude of Jupiter from the ephemeris,.and the difference between.
it and that of the moon, we may infer the longitude of the moon by obfervation : and
from the longitudes fet down for noon and midnight of each day in the nautical Alma-
nac, find the apparent time at Greenwich when the moon had that longitude, which
compared with the apparent time at the fhip, will give the difference of meridians.”
13 | NARRATIVE
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