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20. Explain Borda's circle of repetition; and the method of finding the latitude by the zenith distances of stars near the meridian.
21. The autumnal equinox takes place at 6 in the evening, the moon being full at the same instant, and in her ascending node; the next night the moon rises at the same hour. Required the north latitude of the place.
22. By what methods may the variation of the compass be determined; and to what point does the true north correspond when the variation is 22° 30′ west?
23. Find the latitude of the place in which the longest day contains 16 hours.
24. The plane of a vertical dial is inclined at an angle of 1
45° to the plane of the meridian in a latitude whose sine = √3;
find the position of the substile, the altitude of the stile, and the hour lines.
25. The earth being considered a perfect sphere, prove that at any place the length of a degree of latitude : the length of a degree of longitude :: radius: cosine of latitude.
26. Given the latitudes and longitudes of two places on the earth's surface, to find their distance.
27. Given the latitude of the place and the length of the day, to find the time of the year.
28. What effects are produced by aberration in the apparent places of the moon and the planets ?
29. Given the place of a planet at noon, on March 20th,
on the 22nd in
in Libra 3° 4' 30": on the 21st in 8° 7' 7" :
to find its place
30. If Jupiter and Saturn are in conjunction with one another, and in opposition to the sun, on a given day; and their periodic times are 12 years and 29.5 years respectively; when will they again be in the same position?
31. Given the altitude of the sun, and the breadth of the penumbra which the top of a mountain throws upon a horizontal plane; to find the height of the mountain.
32. Explain the method of deducing the sun's parallax from the transit of an inferior planet.
33. Two places in the same latitude whose difference of 1823 longitude is l, are distant a miles from each other; find their latitude.
34. If % = true zenith distance of a planet, p = its parallax at that distance, and P = horizontal parallax; then
35. The sine of half the angle that measures the duration of sin 9° the shortest twilight cos lat
36. If a comet move in a hyperbola whose semi-axis = a, and eccentricity ae, its place at the end of t" after leaving the perihelion may be determined from the equations,
= tan tan2 (45° + 2).
e tan - log tan
(2) tan (1) tang
where v = true anomaly reckoned from the perihelion, and P = earth's periodic time, her mean distance being 1.
37. If the longitudes of a planet in three different points of its orbit be denoted by a, b, c, and its latitudes at those points by a, ẞ, y; then will
tan ẞ. sin (c-a) = tan a . sin (c—b) + tan y. sin (c—a).
38. If the sun's longitude = c, and the obliquity of the ecliptic, then will the equation of time arising from the obliquity of the ecliptic
= tan2 sin 2c - tan1sin 4c + tan sin 6c —
ad infinitum converted into time.
39. Find the perihelion distance of the comet, moving in the plane of the ecliptic, that stays the longest time within the earth's orbit.
40. If S surface of a portion of the earth ABCD, AB being an arc of the equator, and AC, BD two arcs of circles of latitude; also if AB = c, AC = a, BD = b, then will
41. In consequence of the aberration of light, every star appears to describe an ellipse in the heavens, of which the true place of the star is the centre. Prove this, and find the axes of the ellipse.
42. Given the latitude of a place, find the time of the year when a given star rises at a given hour.
43. Explain the construction, adjustment, and use of the zenith sector.
44. Given the precession in right ascension of a star, find the corresponding change in the angle of position.
45. Find the height of a lunar mountain.
46. Find the time of the sun's passing the vertical wire of a telescope.
47. In what latitude is the angle between 3 and 4 o'clock hour lines, on a horizontal dial, a maximum?
48. In an ellipse find the locus of the intersections of the lines SP and CQ, which cut off the true and eccentric anomalies.
49. If the radius vector and perpendicular on the tangent in any curve described by a revolving body be denoted by r and p; and in the curve of a star's apparent aberration, as seen from this body, by r' and p', then will r : p :: r' : p'.
50. On a given day, in a given latitude, the sun being on the meridian, determine geometrically the angle at which a rod of given length must be inclined to the horizon, that its shadow may be the greatest possible.
51. The altitudes of two stars as they cross the prime vertical are observed, and the difference of their right ascensions is known; find the latitude of the place.
52. State how the sun, planets, and fixed stars are affected by aberration; and shew that the part of the aberration arising S being the
from the motion of the planet varies as
sun, T the earth, and P the planet.
53. If I be the obliquity of the ecliptie, L the sun's longitude, A his right ascension, then
I sin 2L
I sin4L 2 sin 2′′
2' sin 3′′
54. If the latitude of a place be determined by observing 1825 the altitude of the sun at 6 o'clock, and the tabulated declination be affected by a small error, find the corresponding error in latitude.
55. Given the distance of Jupiter from the sun, his radius, and the times of his diurnal and annual revolutions, to compare the aberration of a given star when it passes the meridian of an observer in his equator at mid-day and at mid-night.
56. In a given latitude, find the altitude of the sun on the day of the equinox at 9 in the morning.
57. If v represent the true anomaly of a planet, reckoning from the perihelion, u the eccentric anomaly and e the eccentricity; when e is small, v — u = e sin u, nearly. Required proof.
58. Explain the cause of aberration of light; shew how it is to be measured, and distinguish accurately between the aberration of the fixed stars and the aberration of the planets.
59. Given the latitude of the place and the declination of the sun. Find the time that the sun is above the horizon.
60. Given the time of sunrise and the altitude of the sun when due east on the same day, to find the latitude of the place and the declination of the sun.
61. The N.P.D. of a Aquila being 81° 38′ 25′′, and its observed zenith distance when on the meridian 43° 50′ 45′′, find the latitude of the place; and state the several corrections which must be applied to obtain an accurate result.
62. Shew that a horizontal dial, constructed for north latitude 7, will be a vertical meridional dial for south latitude 90 1.
63. If P be the pole of the heavens, Z the zenith, and S a given star, find when the angle ZSP increases fastest.
64. At what hour on a given night, in a given latitude, will the vertical circle passing through a known star cut the equator in a given angle?
65. Enumerate the arguments by which the diurnal rotation of the earth round its axis and its annual motion round the sun, are established.
66. In any latitude find when the time of the rising of the sun's disk bears the least ratio to the time of its crossing the meridian.
67. If λ be the true latitude of a place, and ✪ the latitude on Mercator's chart constructed to a radius = 1; prove that 2 tan = e☺ — e ̄0.
68. Find the interval between the heliacal rising and setting of a given star, to a spectator in a given latitude.
69. Prove that the semi-axes major and minor and the semilatus rectum of an elliptic orbit are respectively an arithmetic, geometric, and harmonic mean, between the aphelion and perihelion distances.
70. Having given the contemporaneous altitudes of the sun and a known star, on a given day, and also the angular distance between them; find the latitude of the place and the hour of the day.
71. Determine when the sum of the zenith distances of two known stars in a given latitude is a maximum.
72. It is required to graduate a horizontal dial, the style of which is in the meridian, and inclined to the horizon at a given angle, so that on a given day it shall shew the apparent time in a given latitude.
73. There were five Sundays in February 1824; in what year will this happen again?