74. Find when the inclination of the ecliptic to the horizon increases fastest. 75. What probable and adequate cause has been assigned for the secondary planets always turning the same face towards their primaries? 76. If a star whose right ascension is 19° 25′ pass over the meridian 2h 18' of sidereal time before the sun, what is the sun's right ascension when on the meridian? 77. Explain the moon's phases, and why part of the disk is always visible. 78. The times of the sun's rising and setting being calculated for a certain place, what correction is necessary to make them serve for another place not far distant from it? 79. The style of a horizontal dial being bent down, its edge coincides with the 9 o'clock hour-line. For what latitude was it constructed? 80. Supposing the sun to remain above the horizon a given number of days, find the latitude. 81. If L be the length in miles of an arc of a great circle of the earth, D the depression in feet of one extremity of it 2 below a tangent drawn at the other, D = L2 nearly. 3 82. Prove that Brinkley's formula for the mean refraction is reducible to the same form as Bradley's. 83. Given the position of the moon's nodes, and the inclination of her orbit to the ecliptic, to find when her latitude and declination are equal. 84. In a chart on Mercator's projection the length of the meridian from the radius of 30° to that of 60° is to the radius √3+1 of the sphere as the natural logarithm of 3 √3 to 1. 85. Supposing the latitude of a star to be 60°, its longitude 95°, and that of the sun 65°, what is the aberration in longitude? In what sense is 20" 25 the maximum of aberration? 1828 86. On the supposition of a homogeneous atmosphere, the refraction may be expressed by the formula m 1 r = tan Z (2 d (m − 1) (1 + d) 8 being the ratio of the height of the homogeneous atmosphere to the radius of the earth. + √) r), 87. Given the altitudes of two known stars at the same instant of time; required the latitude of the place. How may this problem be solved geometrically on a sphere? 88. Convert 17° 25′ 8′′ into time at the rate of 15° to one hour. 89. Find the azimuth of two known stars which are seen at the same instant in one vertical plane. 90. A known circumpolar star reaches its maximum azimuth at two different places at the same instant; having given the values of the maximum azimuth at the two places, find their latitudes and the difference of longitude. 91. State the principal arguments for the diurnal rotation of the earth round an axis, and its annual motion round the sun. 92. Having given the right ascension and declination of a star, find its latitude and longitude, and adapt the formula to logarithmic computation. 93. What must be the relation of the distances from the sun, of a superior and inferior planet, that their synodical revolutions may be equal? t = 94. Three stars A, B, C are very nearly in a great circle, the angle made by A and C at B being 180 6, where ẞ is small. Shew that if t be the time which elapses between AB and BC being vertical, sin z β cos a cos / 15' % where is the zenith distance and a the azimuth of B, and the latitude of the place. 95. A style projects from the vertex of an upright cone; trace the hour lines on the surface of the cone; and find the time in each day during which the dial will serve. 96. If a body fall to the earth in the time t", the deviation to the east of the point from which it fell will be gat3 cos l; where is the latitude, and a the angle described by the earth in 1". 97. Given three altitudes of a known star observed very near the meridian, and the differences of the times of observation; determine the latitude of the place. 98. Describe the manner in which Bradley discovered aberration, and in which he distinguished nutation from it. sin z p sin P sin 1" 99. If z be the true zenith distance, P the horizontal parallax, p the parallax in seconds, = 100. Having given the latitude of the place, find the day of the year on which the shadow of a given ellipse placed perpendicular to the meridian with its major axis vertical will be an ellipse of half the eccentricity. 101. The interval between the passages of a known circumpolar star through the plane of a vertical instrument with a given azimuth is observed; find the latitude of the place and the area of the spherical surface contained between the vertical circle and the apparent path of the star. 102. Explain the use of observations made by reflexion on the polar star in adjusting a transit instrument. 103. Compare the portion of the surface of the earth illuminated by the sun in perigee with that illuminated in apogee, taking into account the magnitude of the sun. 104. Apply Lagrange's theorem to the determination of the first three terms of the series expressing the true anomaly in terms of the mean, the series ascending by powers of & and the anomaly being measured from the perihelion. ε 105. Given three distances of a planet from the sun, and the corresponding arguments of latitude, to find the place of the perihelion, and the true anomaly at the first observation. 106. Construct a vertical south dial for a given latitude. 1829 107. Shew how a planet, superior or inferior, may have its motion direct or retrograde, or may be stationary. 108. Explain the method of determining the sun's parallax by observations made on the transit of Venus over the sun's disk. 109. Find the aberration of a given star in R.A., in terms of the R. A., declination, obliquity, and sun's longitude. 110. Find the precession in north polar distance. 111. Explain the causes of change of seasons, and of the different lengths of day and night. Shew that the full moon in winter is longer above the horizon than in summer. 112. Explain by what observations the path of the sun among the fixed stars is determined. 113. Shew that the aberration of a star takes place towards a point of the ecliptic 90° before the earth's place, and that it varies as the sine of the angle of the earth's way. 114. Explain the cause of twilight, and shew how its duration may be found from the declination of the sun and the latitude of the place. Find also on what day it is shortest at a given place. 115. The style of a horizontal dial, which is accurately graduated for a given place, is bent through a small given angle d; if a be the hour angle at which there is no error in the time denoted by the dial, find the error in the time denoted at any hour angle h, on a given day; and shew that at six o'clock the error is independent of the latitude of the place, and vanishes at sunset. 116. Explain the use of the astronomical clock, and shew how we may determine whether it goes uniformly at all hours. Also, explain clearly why the mean daily rate determined by the assistance of tables, differs from that determined by direct observations on the transits of stars, and state for what stars this disagreement is perceptible. 117. Explain Flamstead's method of determining the right ascension of the sun, by observations made near the equinoxes; and shew that it will serve to determine the place of the equinox. 118. On a given day and hour, find the azimuth of the ascending point of the ecliptic, and the longitude and altitude of the nonagesimal degree. 119. Find the times of the beginning and end of a lunar eclipse, and the number of digits eclipsed. 120. The first of February of the present year (1829) falls a Sunday. Find generally when this will happen again; and write down all the years in which it will occur during the present century. 121. If at a place between the tropics, c be the zenith distance of a known star, when the ecliptic comes upon the zenith of the place of observation, and the longitude of the earth, when the corresponding aberration in zenith distance vanishes, prove that is determined by the equation sin2 cos c ✔sin (c + λ) sin (c — λ) where L and represent the longitude and latitude of the star. cot (0 - L) sin2 122. If the observed angular distance of two points be a, their observed elevations above the horizon being H and h, and the angular distance reduced to the horizon, shew that 0 sin(a + H - h) sin 1 (a + h cos H cos h 922 - H) = 123. Find when the altitude of a known star increases fastest. 124. Explain fully the method of determining the sun's parallax by the transit of Venus over the sun's disk. 125. Explain accurately what is meant by the equation of time; and shew that it vanishes four times in a year. 126. Construct a vertical south dial, and determine how much of it must be graduated. 127. Having given three geocentric places of a comet, find the corresponding heliocentric and geocentric distances. 128. Find the moon's parallax by observations made out of the plane of the meridian, and shew how the effect of refraction may be avoided. |