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that is, sin (P+P) =

cos a

COS D

....

(7.)

Hence from equa. 6 and 7, we may determine p' and P, the hour angles at the beginning and end of the twilight, D the sun's declination being determined by equa. 3.

We have also COS $ =

sin L
COS D

(8.)

17. Let ST = 2a = zb; and draw PT, then is ZT = 90°, and T is a point in the horizon for the moment when the zenith was in z. The triangle PZT gives (chap. vi. art. 21.) *

• PT D, or PT = 90°- D.

COS FT = cos z sin PZ sin ZT + cos PZ COS ZT = cos z sin pz =tan a tan L cos L tan a sin L= sin D. -Therefore 90° But Ps 90° + D; therefore PT + PS 180° This is another remarkable symptom of the shortest twilight.*

Example.

(9.)

Required the day on which the twilight is shortest, at Woolwich, N. lat. 51° 28′ 40′′, in the year 1816, with its duration and the time of its beginning and end.

First, for the declination, sin D = –
To log tan a.. 9o
Add log sin L..51° 28'

tan a sin L

9.1997125

....

9.8934104.

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Thus the declination is south, and answers to March 2d, and October 11th,

Many other curious particulars connected with this problem are given in Astronomie Théorique et Pratique, par M. Delambre, chap. 14; whence the above investigation has been extracted. Another elegant investigation may be seen in Connaissance des Tems, pour l'an 1818, pp. 382-404.

Next, for the duration, sin (P' — P) =

From log sin a 9°.:
Take log cos L ..51° 28′

Rem. log sin.......... 14° 32′ 41′′..

sin a

COS L

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9.1943324
9.7943613

9-3999711

Hence P P 29° 5′ 22′′, or in time 1h 56m 211s.

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Rem. log sin 95° 31′ 19′′.... 9.9979800

=

....

This value of (P' + P) reduced to time is equivalent to 6h 22m 51, while (PP) is 58 102. The sum of these is P'7" 20m 164, the end of the evening twilight; their difference is P5h 23m 543, the time of the sun setting or the beginning of evening twilight. These respectively taken from 12, leave the times of the begining and end of the morning twilight.

PROBLEM IV.

18. Assuming the error in taking an altitude, it is required to determine the corresponding error in time, in the case of example 4, prob. ii.

E

m

Let EQ in the marginal figure be the equator, P the elevated pole, ab the parallel of de-magaz clination in which the sun or other heavenly body is on the day of observation; and let r be the real and s the apparent place of the body, or rs the error in altitude. Draw ms parallel to the horizon, from z the zenith draw zm, zr, and from P the pole, the meridians, Pmp, Prq, to

pass through m, r, and cut the equator in the points p; q• Then m and r will be the apparent and real places of the body on the parallel of declination, and the arc mr on that circle, or pq on the equator, will measure the angle mer, the corresponding error in time. The triangle msr being, of course, exceedingly small, may be regarded as a rectilinear triangle right angled at s.

Hence, sr: rm:: sin smr: rad.

and mr: pq:: cos qr: rad.

Conseq. multiplying the corresponding terms we have srpq:: sin smr cos qr; rad2

rad

Whence pq=sr

....

(1.)

sin smr cos qr

But zrp smr, srm being the comp. of each; and sin zrr (= sin smr): sin PZ:: sin rzp: sin pr; Conseq. sin smr sin pr= sín smr cos qr sin pz sin rèr. Therefore, by substituting for the denominator in equa. 1, its equivalent in the last, we have

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Cor. Hence, since sin azimuth is a maximum when the body is on the prime vertical, the error in time will then be a minimum, the refraction being accurately allowed for. Wherefore,

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Cor. 2. It is best to deduce the time from an altitude taken when the sun or other body is on or near the prime vertical.

[See, on kindred topics, chap. xi.]

Example.

Suppose that in example 4, prob. 2. the error in altitude was 1', what would be the corresponding error in time?

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PROBLEM V.

19. The right ascensions and declinations of two fixed stars, observed on the same vertical given in position, together with their distance, or the arc of that vertical included between them, being known; it is required to determine the latitude of the place of observation.

E

Z

S

If s, s', in the annexed diagram, be the places of the two stars in the vertical zso, HR the horizon, P the elevated pole, and EQ the equator; then it is evident that the three sides of the triangle PSS' are given, as well as the angle at P. For PS, PS', are the H codeclinations of the stars (known

D

D

by hypothesis), ss' their distance on the vertical given in position, and the angle SPS' the measure of the difference of their right-ascensions. Hence, by the proportionality of the sines of opposite sides and angles, we shall have,

sin ss': sin sPS:: sin s'p: sin rss' = sin psz

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Thus, two angles and a side opposite to one of them, will become known in the triangle rzs, namely, the azimuth, pzs, the angle Psz, and the side ps; whence we shall have

sin PZS: sin PSZ:: sin PS: sin PZ = cos L

sin PS sin PSZ sin Ps sin Ps' sin SPS'

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sin Ps sin Ps' sin SPS' cosec ss' cosec PZS.

20. Cor. 1. It is evident that the hour angle ZPS may be readily found from the same data; and therefore if the sun's place and his right ascension be known, the difference between it and that of the star, s, will become

known; and if from this difference the arc ED be taken, the remainder will give the arc of the equator contained betwixt the meridian of the place and that which passes through the sun, whence the moment of observation may at once be deduced. It is equally obvious that, by pursuing the computation, the altitude of each of the stars at the moment of observation may be found.

21. Cor. 2. If, instead of the azimuth of the two stars, their altitudes were given, we might readily determine the elevation of the pole; for, in that case we should know the three sides of the triangle PSS' with the angle P; and thence the angle Psz between the given sides ps, zs, of the triangle PSZ.

22. Cor. 3. So again, if besides the declinations and right ascensions of the stars observed on the same vertitical, the hour angle zps were given, the latitude might be ascertained; for then, in the triangle Psz, the angles ZPS, ZSP, and the side SP, would be known, and ZP would be determinable by case 3, prob. 3, of oblique spherical triangles.

23. Cor. 4. Hence, from this problem and corollaries the latitude of a place may be determined with great facility; a plumb line serving to show when two stars are in the same vertical, and the altitude being susceptible of being taken with sufficient accuracy for common purposes without employing large instruments. But correct tables of the places of the stars are indispensable.

Example.

Two stars, the right ascension of one of which s is 78° 24', and its declination 27° 25', the right ascension of the other, s', is 104° 52′, and its declination 12° 18′, of the same kind with the former, are distant from each other 28° 30′, on a vertical whose azimuth Ezo is 73° 36. Required the latitude of the place?

Here PS

90°— 27° 25′ — 62° 35′, rs′ = 90° =77° 42',

12° 18'

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