π Τ'. t m sin (1 – 8) , very nearly; l being the latitude of the place, cos o 8 the declination of the sun's centre, and m the ratio of the apparent motions of the centre in declination and right ascension. 219. At a given hour on a given night the moon is observed to rise in the east point; determine the longitude of the node of her orbit, supposing its inclination to the ecliptic known. 220. If during one revolution of a planet the times when it crosses the ecliptic and attains its greatest latitudes be observed, shew how to determine whether its orbit is eccentric, and what is approximately the inclination of its axis major to the line of nodes. If the axis major be perpendicular to the line of nodes, prove that an approximate value of the eccentricity is obtained from the expression where T and t repre 8 sent the respective times of motion on different sides of the ecliptic. 221. Shew that the equation of time attains two maximum and two minimum values in a year, and deterinine approximately the corresponding positions of the sun. Prove that the maximum value, which occurs next after the summer solstice, is much less than that which occurs next before the vernal equinox. 222. A transit instrument is placed with its axis N and S in a horizontal position; find the latitude by observations upon a known star. 223. Shew how the longitude of a place on the earth's surface may be found by observing the distance of the moon from the sun or a fixed star, What other methods are used to determine the longitude, and why is it more difficult to ascertain the longitude than the latitude of a ship? 224. Construct a horizontal dial. Why is the style parallel to the earth's axis? If it deviate from this position by a small given angle, and the time be known when the error is nothing, find the correction for any other time. 225. Find the error with which the hour angle of a heavenly body is affected, when computed from an observed zenith distance, as far as it arises from parallax. Apply it to find the moon's parallax, and thence her distance from the earth. 226. Find the aberration perpendicular to any plane, taking into account the elliptic form of the earth's orbit. 227. State the theorem by which the computation of a triangle on the terrestrial sphere is reduced to that of a plane triangle having sides of the same length, and apply it to solve a geodesic triangle where two sides and the included angle are given. What advantage does the approximate possess over the exact method ? 228. Explain the correction of annual parallax for a fixed star, and investigate its effect on the declination. Shew that its present amount in any direction may be determined from the corresponding formula for aberration three months hence. 229. Find the distance of any point on the earth's surface from its centre in terms of the latitude of that point; and shew that the number of seconds in the angle of the vertical is 180.60.60 e sin 21 nearly. What is the use of this expression? 230. Investigate formulæ for changing a catalogue, in which the stars are registered according to their right ascensions and declinations, into one where they are registered according to their longitudes and latitudes. 231. State the observations by which the changes in the 1835 sun's right ascension and declination are determined. How does it appear that the section of his apparent orbit with the celestial sphere is a great circle? 232. Compare the lengths of the longest day at two given places. What is the least latitude, in which the sun does not set for 24 hours? 233. Explain the nature and cause of aberration. In what plane does it take place? Supposing the right ascension of y Draconis to be 270°, describe the effects of aberration on this star at the equinoxes and solstices, which led Bradley to the discovery of aberration; and illustrate your explanation by a figure. T 234. Determine the coefficient of refraction from observations of circumpolar stars. If the declination of any one star be known correctly, shew how a table of refraction may be formed. 235. Shew from observation that the curve which the earth describes round the sun is an ellipse, and determine its eccentricity 236. Supposing an approximate value of the longitude of any place on the earth's surface to be known, shew how a more correct value may be found from an observed occultation of a fixed star by the moon. How may the possible errors of the lunar tables be obviated ? 237. The earth is touched by two equal conical surfaces, the planes of whose bases coincide with that of the equator; and its surface appears projected upon them to an eye placed in its centre. Shew that by a proper assumption of the form of the cones, the earth's surface may be thus projected on a plane circle; and illustrate the formation of the chart by laying down the position of a place of given latitude and longitude. 238. Determine the position of the place nearest to the north pole, at which the sun rises on a given day at the same instant as at Greenwich. If y, z be the zeniths of the place and Greenwich, and yzx a quadrant of a great circle on the celestial sphere, the projection of the locus of x on the horizon of is an arc of an ellipse, whose eccentricity = cos. lat. of Greenwich. 239. Explain the method by which the parallax of a fixed star has been attempted to be found by observations on the fixed double stars; and investigate an equation for determining the times of the year at which the observations on a given star for this purpose may be most advantageously made. If at one of these times u be the difference of the longitudes of the sun and star, the latitude of the latter, 8 the angular distance between the two stars, and a the change of position of the line joining them in the course of half a year, shew that the parallax of the greater star = a sin 8v1 + cos2u.cot?). 240. If I be the latitude of a place between the tropics, a and 8 the sun's right ascension and declination; the times when the ecliptic is vertical are determined from the equation h = a + sin-1 (sin a tan l cot 8). 241. The extremity of the shadow intercepted by a line drawn in the dial plane perpendicular to the substyle of a horizontal or vertical south dial, will move uniformly round some fixed point A in the substyle. If the hour lines be graduated from this property, and a small error be committed in assuming the position of the point A ; the error of the time indicated by the dial at the time t will be 2a sin 30t, where a denotes the error at one o'clock. 242. At midnight, two known stars appear in their real verticals, one at the vernal, the other at the autumnal equinox; shew that their azimuths are supplementary; determine also the latitude of the place and the obliquity of the ecliptic. 243. The altitude of a star on the prime vertical is observed with a sextant whose arc is not exactly 60°, but which is accurately subdivided ; find the error in the arc from observing the difference between the sidereal time deduced from the altitude and the real sidereal time. 244. Having given the heliocentric, find the geocentric place of a planet at any time. How is the time in which a planet is in its node determined ? 245. Having given the lengths of two degrees in the same meridian at different given latitudes, shew how the earth's form and magnitude may be deduced. By what observations is it known (1) that the arc measured is exactly one degree; (2) that it lies in the same meridian ? 246. Explain the method of setting the sidereal clock. A star rises at 5h. 47m. 568. sidereal time, find the mean solar time, supposing the mean sun's right ascension when last on the meridian to have been lh. 47m. 6$. 247. Find the apparent differences of the latitudes and longitudes of the sun and moon at any assigned time during a solar eclipse. 248. If i be the inclination of a plane to the horizon, and a the inclination of a line in it to the intersection of the plane with the horizontal plane; shew that the inclination of the line to the horizon is found from the equation sin ( = sin i. sin a. 1836 249. Explain the use and construction of a vernier ; and if n divisions of the instrument are equal to n + 1 of the vernier, and the mth divisions coincide, determine the true reading off. 250. Determine the time corresponding to a given true anomaly in a very eccentric ellipse; and having given the anomaly in a parabolic orbit, find the anomaly in a very eccentric ellipse, the time and perihelion distance being the same in both orbits. 251. Find the duration of a lunar eclipse. On what account are eclipses of the moon more frequent at a given place than those of the sun. 252. What evidence have we that the earth revolves about the sun, and rotates about its own axis? 253. Describe and explain the phenomena of day and night throughout the year in our latitude. How must the explanation be modified for places near the poles ? 254. How was the motion of light discovered, and by what remarkable astronomical phenomenon was the discovery confirmed? What is parallax ? Having given the horizontal parallax of a heavenly body at the equator, find the parallax of the same body at any other place. 255. Find the latitude of a place by observing three altitudes of an unknown star near the meridian. What are the chief advantages of this method ? 256. Shew how to find the time by an observed altitude of a known star, when east of the meridian. Prove that the error in time from a given error in altitude is less the nearer the star is to the prime vertical at the instant of observation. 257. Explain the terms “solar and lunar ecliptic limits.” Shew that there may be seven eclipses in the course of a year. What effect is produced by the earth's atmosphere in lunar eclipses ? 258. When that part (E) of the equation of time, which arises from the obliquity (w) of the ecliptic, is a maximum, sin E = tan” W |