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TO COMPUTE THE EQUATION OF TIME.

The equation of time is the difference between mean and apparent noon, and consists of two parts, one arising from the obliquity of the ecliptic, and the other from the sun's unequal motion in his elliptic orbit, and may be computed in the following man

ner.

The difference between the mean and true anomalies, or the equation of the center converted into time, is the first part of it. With the sun's longitude and obliquity of the ecliptic compute the sun's right ascension, and the difference between the longitude and right ascension when turned into time, gives the other part. Then the sum or difference of these will give the whole equation.

A solar day is the interval between two successive transits of the sun over the meridian, which is always more than a revolution of the earth on its axis, and at a mean rate is measured by 360° 59′ 8′′; though sometimes more and sometimes less, because of the sun's unequal motion in right ascension. But a sidereal day is the interval between two successive transits of the same star over the meridian, and is always the same, being measured by 360°, and is equal to 23h 56′ 4′′ of a mean solar day: because the difference between a mean solar day, and a sidereal day, viz. 59' 8", when converted into time, gives 3′ 56′′ of time; which shows, that if a star come to the meridian at any hour, it will come to the meridian again 3′ 56′′ before the same hour on the next day, by a clock which measures a mean solar day by 24 hours. The true and mean solar days are never equal, but when the sun's motion in right ascension is equal to 59' 8"; which happens

about Feb. 11, May 14, July 26, and November 1. The accumulation of these differences produces the equation of time.

TO FIND A TRUE MERIDIAN LINE.

This may be done in various ways, some of which may be accomplished with more accuracy than others. Two of the best I shall now mention. Having a clock well regulated to mean time, and a transit telescope, let the telescope be placed in the meridian, as nearly as can be guessed at, by the pole star; and observe the transits of two stars, one to the north and the other to the south, whose difference of right ascension is accurately known, and does not exceed a quarter of an hour. Now, if the difference of the observed times of passing, be the same with the difference of right ascensions, the telescope is in the meridian. But if the northern star pass the meridian first, as will be the case with Ursa minoris, and ẞ Libra, and the interval of observed time be greater than the difference of right ascensions, the telescope verges towards the east of the north meridian, and thereby prolongs the observed interval. If therefore in the next revolution of the northern star, it be kept on the middle wire of the telescope for half the number of seconds, by which the observed interval exceeded the true difference of right ascensions, the telescope following the star for that time, and then be suffered to remain in that position, it will now be exactly in the meridian.

But if the observed interval be less than the calculated difference of right ascensions, the telescope verges towards the west of the true meridian; and therefore the southern star, in the next revolution, must be kept on the middle wire of the telescope for half the number of seconds, by which the calculated difference of right

ascensions exceeds the observed interval of their pas sage.

OTHERWISE THUS.

At a season of the year when the nights are more than twelve hours long, choose a northern star that does not set, and observe its passage both above and below the pole, across the middle wire of the telescope, placed nearly in the meridian, noting the times by the clock. Now if the interval of time between the passages of the star above and below the pole be equal to half a revolution of the stars, as measured by the clock, the telescope is precisely in the meridian. But if the star below the pole come later to the telescope than half a revolution of the stars, after it has passed above the pole, the telescope verges to the east of the north meridian: and if the contrary, it verges towards the west. The error of the telescope may then be calculated and corrected by this

RULE.

To the proportional or logistical logarithm of half the difference between half a revolution of the star and the interval of the two observations, add the logarithm cosine of the star's polar distance, the logarithm cosine of the latitude of the place, and the logarithm secant of the star's altitude; and the sum, rejecting 30 from the index, will be the proportional or logistical logarithm of the time between the observation of the star above or below the pole, and the true time of its passage over the meridian.

EXAMPLE.

Suppose Ursa minoris, whose polar distance is 14° 35′ 30′′, and whose altitude above the pole, at Philadelphia, 54° 33', and altitude when on the meridian below it=25° 22', was observed, at Philadel

phia, whose latitude is 39° 57' 30", to pass through the telescope at 9h 0' 0" by the clock, which loses 4" of solar time per day; and again to pass under the pole through the telescope, remaining in the same vertical position, at 9h 3' 16"; to find the deviation of the telescope from the meridian.

As the clock loses 4" per day, instead of 12 hours, it counts for half a revolution only 11h 59′ 58′′ To which add the time of the star's passing above the pole, viz.

the sum would be the time of its passing under the pole, viz.

had the telescope been right; but it was observed actually to pass at P. at

900

8 59 58

9 3 16

The telescope therefore verges to the eastward of the meridian, because the star passes 3′ 18′′ later than half

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sum of these log. abating 30 from ind. 2.1448=1′17′′.4

error in time of telescope above the pole.

For the error of the telescope by the observation

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If the star, therefore, in its next revolution above the pole, be followed by the telescope for 1' 17".4 of time, and the star be kept on the wire during that time, the telescope will then be in the meridian; in which a mark should be fixed, so as to regain the direction at pleasure. If the telescope had verged too much to the westward, the star must be followed in its passage under the pole, during the time of the error of position under the pole.

DEMONSTRATION.

Let Z* be the zenith of the place of observation, P the pole, and ZAB the position of the telescope near to the meridian ZPC; A, B, the places of the circumpolar star observed above and below the pole. As the telescope is supposed to be placed nearly in the meridian, the angle PZA is but small, as is also the angle ZPA, which measures the difference of time between that of the star's passing through the telescope, and of its passing over the meridian above the pole. For the same reason, AB may be considered as equal to 2AP. The supplement of the angle APB, is the difference between half a revolution of the stars measured by the clock, and the interval of time between the star's passing the telescope above and below the pole, which is to be found.

As S, ABS, 2AP: S, APB, :: S, AP : (S, PBA S,APBXS,AP S,APBx Sec. AP; or PAB=)S,PAZ=

S, 2AP

2

because half the secant of an arch, is as the sine of the arch divided by the sine of double the arch. In the

S,APB

triangle APZ, As I,PZ: (S,PAZ=)

xSec.AP

2

* See Plate 15. Fig. 4.

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