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1830

129. Find the length of the longest day in latitude 52° 13', the obliquity of the ecliptic being 23° 28′.

Having given 10.1105786 =

9.6376106 =
9.7481230=

9.7483099 =

log tan 52° 13',

log tan 23° 28',
log cos 55° 57',
log cos 55° 56'.

130. Explain what is meant by the error of the line of Collimation, and shew how it may be avoided.

131. Having given the sidereal time of any phenomenon, find the corresponding mean solar time.

132. Find the latitude of the place of observation, from two equal altitudes of the sun before and after noon, and the time between.

133. Determine the precession in north polar distance and right ascension, and shew when it is additive and when subtractive.

134. An ellipse may be constructed so that if any abscissa be taken to represent the aberration in longitude of a given star, the corresponding ordinate will represent the aberration in latitude, coordinates being measured from the centre along the axes; prove this and determine the axes.

2

135. Two planets P1, P2 revolve in circular orbits at the distances 7, 7, from the sun, and when they appear stationary to one another cot P2's elongation seen from P1 = 1⁄2 tan 0; shew that TI

2

0

tan tan 0.
2

T2

136. Ax + By + Cz = 0

8}

A'x + B'y + C'z = 0)

in which two planets move. Apply them to find the inclination of the orbits to one another, in terms of their inclinations to the ecliptic and of the longitudes of their ascending nodes, the ecliptic being in the plane of x and y.

are the equations to the planes

137. Explain fully the equation of time, and shew at what seasons that part of it arising from the obliquity of the ecliptic is positive, and at what seasons negative.

138. Shew that the inclination of the ecliptic to the horizon is a minimum when Aries rises, and a maximum when it sets; and explain the phenomenon of the harvest moon.

139. Having given the variation of the obliquity of the ecliptic, find the corresponding variations in right ascension and declination.

140. Determine the latitude of the place of observation from observing the times of the rising of two known stars.

141. Find sin a from the equation

sin x cos x + a sin2x = b,

and shew its use in the solution of the following problem: to determine how much the azimuth of a known star on the horizon is affected by refraction.

142. Find the longitude of the perihelion and the time of the earth's passing through it.

143. Find the sun's right ascension by Flamstead's method. Why must the observations be made near an equinox?

144. Explain Mercator's projection of the sphere, and find the length of the projection of an arc of the meridian included between the latitudes of 30° and 60°.

145. Find the two parts of solar nutation, and prove that they are connected by the equation to an ellipse, the axes of which are in the ratio of cos I : 1, where I is the obliquity of the ecliptic.

146. Having given the length of a degree of latitude, andalso the length of a degree in a direction perpendicular to the meridian, in a given latitude; find the ellipticity of the earth.

147. Shew how to determine whether a planet is a superior or an inferior one; and having given the synodic period of a planet and the length of a year, find the planet's period.

148. Construct a horizontal dial, and find the limits beyond which it is unnecessary to graduate it.

149. Determine the circular orbit of a planet from two observations.

1831

150. In a given latitude find the sun's azimuth, his declination and the time of the day being given; and adapt the trigonometrical formula to logarithmic computation.

151. State the arguments from which it is inferred that the earth revolves round its axis and round the sun.

152. Shew how to draw a meridian line by observing the shadow of a vertical gnomon on a horizontal plane, and correct for the change in the sun's declination between the observations.

153. Explain the nature of the five astronomical corrections : refraction, parallax, aberration, precession, and nutation.

154. The longitude of a Arietis is 35° 4′ 41′′, its north polar distance is 67° 25' 1".7, and the obliquity of the ecliptic is 23° 27' 46'.3; find its angle of position.

log cos 35° 4' 41"

log sin 23 27

log sin 67 25
log sin 20 39

= 9.9129496,

46.3

9.6000517,

1.7

9.9653546,

52.3 = 9.5476467.

155. When the vertical plane in which a transit instrument moves, nearly coincides with the meridian, to find the deviation.

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=

nt = u e sin u,

=

=

Vito

e

U

tan
2'

the former expressing the relation between the eccentric and mean anomaly, and the latter that between the eccentric and true anomaly.

157. What are the sidereal, the tropical, and the anomalistic years? Which is longest, and which shortest?

158. Determine the relative positions of the earth and an inferior planet, when the latter appears stationary.

159. In a given latitude a vertical rod is placed at a given distance from an east and west wall, so as to cast a portion of

its shadow upon it; find the equation to the extremity of the shadow traced upon the wall on a given day. Shew what the equation becomes when the sun is in the equator, and the latitude of the place 45°,

160. Having given u。, u, uga, three values of a function near its maximum, observed at times 0, a, 2a; find the time when the function will be a maximum. And if the declinations of the sun at noon on three successive days were 23°. 27′, 23°. 27.9, 23°. 27'. 6, find when the declination was greatest.

161. If the earth be an oblate spheroid, and from any point Q above it perpendiculars QM, QN be drawn to the axis and equator respectively, intersecting a meridian in P and p; and if tangents PT, pt to this meridian meet the axis and equator in T and t, and if the straight line which joins T and t cuts the meridian in E and F, E and F are the extreme points of the meridian visible from Q.

162. If normals be drawn at every point of the rhumb line, find the locus of their intersection with the equator, the earth being considered an oblate spheroid.

163. Having given the latitudes of two places on the earth's surface, one of which is N. E. of the other, find the difference of their longitudes and their distance from each other, considering the earth a sphere.

b

164. Express the radius of curvature of the meridian in a spheroid of small ellipticity in terms of the latitude.

If X, X' be the latitudes of two stations on the same meridian, prove that the length of the arc included between them is 3e + 2-2

e

sin (X-X)

(x' — x) {1

cos (λ' + λ).

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where

165. Investigate the precession in N.P.D and R. A., and shew when they are additive and when subtractive.

166. Explain the method of finding the longitude of a place by the observed distance of the moon from the sun or a fixed star, and deduce formulæ adapted to logarithmic computation.

167. Shew how the time, duration, and magnitude of a lunar eclipse may be computed.

168. If be the angle which the normal to any point in an ellipse makes with the axis major, the length of the normal b2

a (1-e2 sin2 A)+'

169. Distinguish between a sidereal, a solar, and a mean solar day; and define the equation of time, stating the two causes from which it arises.

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170. Construct a horizontal dial for a given latitude, and determine the limits beyond which it is unnecessary to graduate it.

1832 171. Explain the causes of different lengths of days at different seasons of the year. Shew that the duration of the longest day is greater, and that of the shortest day less as the latitude increases.

172. Explain the effects of parallax; and obtain the value of parallax at a given altitude in terms of the horizontal parallax.

173. Given the latitudes of two places, and their difference of longitude; determine the inclination of their horizons, and the day of the year on which the sun sets to both places at the same instant.

ι

174. Explain the Gregorian correction of the calendar. The length of the mean tropical year being 365.242264 days, in how many years will the error of this correction amount to a day?

175. Determine the relation and the longitude of the sun. a = tan2

w

2. sin 21-
2་

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between the right ascension a, Shew also that

tan1

W

2

sin 4/+

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where w is the obliquity.

176. At what time of the day will a star, of which the right ascension is 270°, pass the meridian at the time of the equinoxes? How will its place be affected by aberration? How was it shewn that the effects of aberration did not arise from a change in the position of the earth's axis?

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