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8. Given the latitudes and longitudes of three places on the earth's surface, and also the radius of the earth; show how to find the area of the spherical triangle formed by arcs of great circles passing through the places.

9. The distance between Paris and Berlin (that is, the arc of a great circle between these places) is equal to 472 geographical miles. The latitude of Paris is 48° 50′ 13′′; that of Berlin, 52° 30' 16". at Berlin?

When it is noon at Paris what time is it

NOTE. Owing to the apparent motion of the sun, the local time over the earth's surface at any instant varies at the rate of one hour for 15° of longitude; and the more easterly the place, the later the local time.

10. The altitude of the pole being 45°, I see a star on the horizon and observe its azimuth to be 45°; find its polar distance.

11. Given the latitude 7 of the observer, and the declination d of the sun; find the local time (apparent solar time) of sunrise and sunset, and also the azimuth of the sun at these times (refraction being neglected). When and where does the sun rise on the longest day of the year (at which time d +23°27′) in Boston (l= 42° 21'), and what is the length of the day from sunrise to sunset? Also, find when and where the sun rises in Boston on the shortest day of the year (when d=23° 27'), and the length of this day.

12. When is the solution of the problem in Example 11 impossible, and for what places is the solution impossible?

13. Given the latitude of a place and the sun's declination; find his altitude and azimuth at 6 o'clock A.M. (neglecting refraction). Compute the results for the longest day of the year at Munich (1=48° 9′).

14. How does the altitude of the sun at 6 A.M. on a given day change as we go from the equator to the pole? At what

time of the year is it a maximum at a given place? (Given sin h sin / sin d.)

15. Given the latitude of a place north of the equator, and the declination of the sun; find the time of day when the sun bears due east and due west. Compute the results for the longest day at St. Petersburg (1 = 59° 56').

16. Apply the general result in Example 15 (cos t = cot l tand) to the case when the days and nights are equal in length (that is, when d=0°). Why can the sun in summer never be due east before 6 A.M., or due west after 6 P.M.? How does the time of bearing due east and due west change with the declination of the sun? Apply the general result to the cases where <d and l = d. What does it become at the north pole?

17. Given the sun's declination and his altitude when he bears due east; find the latitude of the observer.

18. At a point O in a horizontal plane MN a staff OA is fixed, so that its angle of inclination AOB with the plane is equal to the latitude of the place, 51° 30' N., and the direction OB is due north. What angle will OB make with the shadow of OA on the plane, at 1 P.M., when the sun is on the equinoctial?

19. What is the direction of a wall in latitude 52° 30′ N. which casts no shadow at 6 A.M. on the longest day of the year?

20. At a certain place the sun is observed to rise exactly in the north-east point on the longest day of the year; find the latitude of the place.

21. Find the latitude of the place at which the sun sets at 10 o'clock on the longest day.

22. To what does the general formula for the hour angle, in § 69, reduce when (i.) h=0°, (ii.) 7 =0° and d=0°, (iii.) l or d = 90° ?

23. What does the general formula for the azimuth of a celestial body, in § 70, become when t= 90° :

90° 6 hours?

90°, lead to the

24. Show that the formulas of § 71, if t equation sin sin hesed; and that if d=0°, they lead to the equation cos sin h sect.

25. Given latitude of place 52° 30' 16", declination of star 38°, its hour angle 28° 17' 15"; find its altitude.

26. Given latitude of place 51° 19' 20", polar distance of star 67° 59' 5", its hour angle 15° 8' 12"; find its altitude and its azimuth.

27. Given the declination of a star 7° 54', its altitude 22° 45' 12", its azimuth 129° 45' 37"; find its hour angle and the latitude of the observer.

28. Given the longitude v of the sun, and the obliquity of the ecliptic e 23° 27'; find the declination d, and the right ascension r.

29. Given the obliquity of the ecliptic e=23° 27', the latitude of a star 51°, its longitude 315°; find its declination and its right ascension.

30. Given the latitude of place 44° 50' 14", the azimuth of a star 138° 58′ 43′′, and its hour angle 20°; find its declination.

31. Given latitude of place 51° 31' 48", altitude of sun west of the meridian 35° 14' 27", its declination +21° 27'; find the local apparent time.

32. Given the latitude of a place 7, the polar distance p of a star, and its altitude h; find its azimuth a.

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20. sin 4+ sin B = 2 sin (A+B) cos 1⁄2 (A — B).

21. sin A-sin B=2 cos(A+B) sin 1⁄2 (A — B).

22.

cos A+ cos B = 2 cos 1⁄2 (A + B) cos 1⁄2 (A — B).

23. cos A-cos B-2 sin (A+B) sin (4B). 1⁄2

- § 31.

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