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177. From an equinox to a solstice the daily increase of the sun's declination is continually diminished.

178. If the earth's orbit were a circle, shew that the whole duration of daylight at every place on the earth's surface in the course of a year would be the same.

Taking into account the eccentricity and position of the earth's orbit, is there more daylight on the north or south side of the equator?

179. Supposing the eccentricity of the earth's orbit to be equal to cos 89° 2′, and the perihelion at the winter solstice; shew that the summer half of the year is longer than the winter half by 7 days 20 hours nearly.

180. Feb. 6, 1824. Moon's A. R. at Oh was 23° 21′ 55′′,

12. . 29 37 9,

Feb. 6,
Feb. 7,
Determine her A.R. on Feb. 6, at 15h.

0. . 36 5 56.

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181. Having given the error in the observed azimuth of a heavenly body, find the corresponding error in its declination. In measuring an arc of the meridian, shew that the chain of triangles should be near the meridian.

182. Having given the equation between the mean and eccentric anomalies nt = u-e sin u, and that between the true and eccentric

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apply Lagrange's theorem to find a series for v in terms of nt as far as e3.

183. Explain what is meant by the equation of time; and find when that part of it which is caused by the obliquity is additive or subtractive, and when it is greatest.

184. Investigate the nutation in right ascension of a given star: also express the nutation in terms of the maximum nutation, the longitude of the moon's ascending node, and the longitude of the moon's ascending node corresponding to the maximum nutation.

185. Having given the lengths of two arcs of the meridian, and the latitude of their extremities, find the compression of

1833

the earth. If the arcs be on the same side of the meridian, the operation is more accurate in proportion as they are more distant from each other.

186. In orbits of small eccentricity, the greatest equation of the centre is nearly twice the eccentricity.

187. If R be the refraction of a given apparent zenith distance, when the temperature is 50°, and the barometer 29.6 inches, and r the refraction for the same zenith distance, when the temperature is 50° + t°, and the barometer 6 inches; then 1-BR nearly; the elastic force of the air r = 29.6 + at

increasing a parts of the whole, and the mercury expanding B parts of the whole, for each degree of temperature above 50°.

188. Having given the perihelion distance D of a comet in a parabolic orbit, find an expression for the time through any angle of true anomaly; and having given corresponding tabulated values of v and t in an orbit whose perihelion distance is 1, find corresponding values of v and t in any other orbit.

189. Enumerate the elements of a planet's orbit. What does each determine?

Find the node of a planet's orbit from observations made on the planet near its node?

190. Explain Mercator's projection of the sphere, and find the length of an arc of the meridian on this projection, supposing the earth spherical.

191. Explain the cause of twilight; and determine its duration at a given place on a given day.

192. Find the relation between the mean and eccentric anomalies, and that between the eccentric and true anomalies.

193. Explain the moon's phases, and state the cause which renders the dark part visible near new-moon.

194. State the three laws of planetary motion discovered by Kepler, and the nature of the observations by which they were established. Shew that in the conjunction of Jupiter and Saturn the disturbance which the former produces on the orbit

of the latter is greater than the effect of the latter on that of the former.

195. By what observations is it shewn that the sun's apparent annual motion is in a great circle? Shew how the obli

quity of the ecliptic is determined.

196. Explain the construction of the common horizontal dial, and draw the last hour lines in a given latitude.

197. Determine the position of the ecliptic with respect to the meridian and horizon of a given place, at any time. In what astronomical investigations are the quantities determined above required?

198. Determine the longitude by observing the increase of the moon's right ascension in the interval between passing the meridian of Greenwich and that of the place of observation.

199. Find the moon's parallax by observations made not in the plane of the meridian; and shew how the correction for refraction may be avoided by observing the zenith distance of a star nearly in contact with the moon.

200. Express the latitude and longitude of a star in terms of its observed altitude and azimuth, the observations being made at a given place when the first point of Aries was on the meridian.

201. Determine the points of its orbit between which the earth must be situated when a given superior planet appears stationary; the orbits of both earth and planet being supposed circular, and in planes inclined to each other.

202. Find the latitude from observing the angular distance of the extreme points of the horizon in which the sun appears at rising in the course of a year; and if a, ß denote the distances of those points from the point in which the sun rises when the declination is 8, prove that sine of the obliquity

sin (a + B)
sin(a)'

= sin &.

203. Let two heliocentric distances of a comet revolving in a hyperbolic orbit be denoted by p, p'; and the length of the

line joining the two positions of the comet by c; and let p,p′ be two angles, such that

--

p + p' + c = 2a (sec p′ − 1), p+ p c = 2a (sec p − 1), 2a being the major axis; prove that (taking the earth's mean distance and period for the units of distance and time) the time of moving between the two positions

{tan + tan

o

log, (tan + tan p') (sec + sec p')}. 204 If D be the apparent diameter of the sun, ✪ the altitude of its centre, and s1, s, the respective lengths of the pure shadow and penumbra cast by a vertical rod upon a horizontal plane, prove that

1 (sin 20

2 sin D

$1 $2

−1}.

205. Find the aberration of a given star in right ascension. In what respect was y Draconis peculiarly fitted for discovering the effects of aberration, at Greenwich? How did Bradley separate the effects of aberration from those of nutation and precession?

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206. Shew how to find the inclination of the sun's equator to the ecliptic, and the time of the sun's rotation. On what grounds is it probable that the sun, in addition to his rotatory motion, has a motion of translation?

207. To determine the latitude from zenith distances of a known star observed very near the meridian.

208. Point out all the circumstances in which solar and lunar eclipses differ; and calculate the circumstances of a solar eclipse at a given place.

209. To find Jupiter's distance from the sun by an observed eclipse of a satellite. Explain the method of determining the longitude by observations of those eclipses.

210. Having given the inclination to the primitive of a great circle of the sphere, find the radius and position of the centre of its stereographic projection; and shew that the centres of the projections of the meridians upon the horizon of any place lie in a straight line.

211. Express the radius of curvature at any point of the elliptic meridian in terms of the latitude. Mention the various methods that have been employed to determine the figure and dimensions of the earth. If the length of a degree be 69 miles, what will be the length of the earth's radius supposing it a sphere?

212. By what observations does it appear that the stars 1834 describe parallel circles with a uniform angular motion, all completing their revolutions in the same time, at a distance compared with which the dimensions of the earth are evanescent?

213. Describe the construction and adjustments of the mural circle; and the nature and objects of the observations made with it.

214. Find the latitude and hour angle from two altitudes of the sun and the time between, neglecting the change of declination in the interval. At what time of the year will the result thus obtained be the most accurate?

215. Explain the principle of Mercator's method of projection; and find in it the length of an arc of the meridian, supposing the earth to be a sphere.

216. Determine the precession in declination of a given star whose right ascension is less than 90°. How is the mean value of the whole precession determined from observation? If its amount be determined from the observed annual precession of a star in declination, why is the result different from that which would be obtained from the observed annual precession in right ascension?

217. Explain the phases of an opaque heavenly body illuminated by the sun's rays. What is the cause of the occasional gibbous appearance of Mars? and why is not the same perceived in the other superior planets? At what point of his orbit is this appearance most remarkable?

218. Why are the sun's transit and his culmination not contemporaneous events? Prove that in the interval between them, his centre describes an hour angle whose sine equals

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