13. cos C =- cos A cos B + sin A sin B cos c.

14. cot a sin b = cot A sin C + cos C cos b.

15. cot a sin c = cot A sin B+ cos B cos c.

16. cot b sin a cot B sin C+ cos C cos a.

=

17. cot b sin c = cot B sin A + cos A cos c.
cot C sin B+ cos B cos a.
= cot C sin A + cos A cos b.

18. cot c sin a
19. cot c sin b
20. sin a cos B = cos b sin c

21. sin a cos C sin b cos c

=

=

sin b cos c cos A (Art. 194) cos b sin c cos A.

sin a cos c cos B.

sin (s — b) sin (s

sin b sin c

sin s sin (sa).

sin b sin c

sin (s - b) sin (s—c).
sin s sin (s - a)

2√sin s sin (s — a) sin (s — b) sin (s – c)

2n

sin b sin c

where

n = √sin s sin (s — a) sin (s — b) sin (s — c).

2√-cos S cos (S-A) cos (S-B) cos(S-C)

[sin(s-a)+sin(s-b)+sin(s-c)-sin s]

46. tan E = √tan stan (s—a) tan (s—b) tan † (s—c).

EXAMPLES.

1. Find the time of sunrise at a place whose latitude is
42° 33' N., when the sun's declination is 13° 28' N.

Ans. 5h 9m 139.-

2. Find the time of sunset at Cincinnati, lat. 39° 6' N.,
when the sun's declination is 15° 56' S.
Ans. 5h 6m.

3. Find the time of sunrise at lat. 40° 43' 48" N., in the
longest day in the year, the sun's greatest declination being
23° 27' N.
Ans. 4h 32m 16.4.

4. Find the time of sunrise at Boston, lat. 42° 21' N.,
when the sun's declination is 8° 47' S.
Ans. 6h 14m.

5. Find the length of the longest day at lat. 42° 16′ 48′′.3
N., the sun's greatest declination being 23° 27′ N.

Ans. 15h 5m 50o.

6. Find the length of the shortest day at New Bruns-
wick, N.J., lat. 40° 29′ 52'.7 N., the sun's greatest declina-
tion being 23° 27' S.

7. Find the hour angle and azimuth of Antares, declina-
tion 26° 6' S., when it sets to an observer at Philadelphia,
lat. 39° 57' N.
Ans. 4h 23m 5.7; S. 54° 58′ 44′′ W.

8. Find the hour angle and azimuth of the Nebula of
Andromeda, declination 40° 35' N., when it rises to an ob-
server at New Brunswick, N.J., lat. 40° 29′ 52".7 N.

9. Find the azimuth and altitude of Regulus, declination
16° 13' N., to an observer at New York, lat. 40° 42' N., when
the star is three hours east of the meridian.
Ans. Azimuth = S. 71° 12′ 30′′ E.; Altitude

= : 44° 10' 33".

10. Find the azimuth and altitude of Fomalhaut, dec-
lination 30° 25' S., to an observer in lat. 42° 22' N., when the
star is 2h 5m 36s east of the meridian.

Ans. Azimuth = S. 27° 18′ 40′′ E.; Altitude

= 11° 41' 37".

in lat.

11. Find the azimuth and altitude of a star to an observer
39° 57' N., when the hour angle of the star is
5h 17m 40 east, and the declination is 62° 33' N.

12. Find the hour angle (t) and declination (8) of a star
to an observer in lat. 40° 36′ 23′′.9 N., when the azimuth of
the star is 80° 23' 4".47, and the altitude is 47° 15' 18".3.
Ans. t = 46° 40′ 4′′.53; 8 = 23° 4' 24".33.

13. Find the distance between Regulus and Antares, the
right ascensions being 10h 0m 29.11 and 16h 20m 208.35, and
the polar distances 77° 18′ 41′′.4 and 116° 5' 55".5.

Ans. 99° 55' 44".9.

14. Find the distance between the sun and moon when
the right ascensions are 12h 39m 38.22 and 6h 55m 323.73, and
the declinations 9° 23' 16".7 S. and 22° 50′ 21′′.9 N.

Ans. 89° 52′ 55′′.5.

15. Find the shortest distance on the earth's surface, in
miles, from New York, lat. 40° 42′ 44′′ N., long. 74° 0′ 24" W.,
to San Francisco, lat. 37° 48' N., long. 122° 23' W.

Ans. 2562 miles.

16. Find the shortest distance on the earth's surface
from San Francisco, lat. 37° 48' N., long. 122° 23′ W., to
Port Jackson, lat. 33° 51' S., long. 151° 19' E.

Ans. 6444 nautical miles.

17. Given the right ascension of a star 10h 1m 98.34, and
its declination 12° 37' 36".8 N.; to find its latitude and longi-
tude, the obliquity of the ecliptic being 23° 27' 19".45.

Ans. Latitude =

; Longitude =

18. Given the obliquity of the ecliptic w, and the sun's
longitude λ; to find his right ascension & and declination 8.
Ans. tan a cos o tan λ; sin 8 sin o sin X.

=

19. Given the obliquity of the ecliptic 23° 27' 18".5, and the sun's longitude 59° 40' 1".6; to find his right ascension (a), and declination (8).

Ans. a 3h 49m 52.62; &= 20° 5′ 33′′.9 N.

20. Given the sun's declination 16° 0′ 56′′.4 N., and the obliquity of the ecliptic 23° 27' 18".2; to find his right ascension (a), and longitude (λ).

Ans. a 9h 14m 193.2; λ = = 136° 7' 6".5.

21. Given the sun's right ascension 14h 8m 19.06, and the obliquity of the ecliptic 23° 27' 17".8; to find his longitude (A), and declination (8).

Ans. λ= 214° 20' 34".7; 8 12° 58' 34".4 S.

=

22. Given the sun's longitude 280° 23′ 52".3, and his declination 23° 2' 52".2 S.; to find his right ascension (a). Ans. a 18h 45m 148.7.

23. In latitude 45° N., prove that the shadow at noon of a vertical object is three times as long when the sun's declination is 15° S. as when it is 15° N.

24. Given the azimuth of the sun at setting, and also at 6 o'clock; find the sun's declination, and the latitude.

25. If the sun's declination be 15° N., and length of day four hours, prove tan & sin 60° tan 75°.

=

26. Given the sun's declination and the latitude; show how to find the time. when he is due east.

27. If the sun rise northeast in latitude 4, prove that cot hour angle at sunrise - sin .

28. Given the latitudes and longitudes of two places; find the sun's declination when he is on the horizon of both at the same instant.

29. Given the sun's declination 8, his altitude h at 6 o'clock, and his altitude h' when due east; prove sin2 8 sin h sin h'.

=