If l,d be a star's latitude and declination, its distance (d) from the sun, at the moment of his crossing the equator, may be found from sin2 d. sin2 = sin2 / 2 co: o sin / sin & + sin2 &; prove this equation, and prepare it for calculation by a table of logarithms. 259. In 365, 5h, 48m, the sun's longitude is increased by 360°; what is his mean daily motion? 260. Having given the sun's altitude and azimuth, determine the points where an arch of given colour in the primary rainbow meets the horizon. At a place of given latitude determine the season of the year, during which it is impossible to see a primary rainbow at midday; and on a given day, during that period, for what time before and after noon this is the case. 261. At a place of given latitude determine the nearest approach of the ascending point of the ecliptic to the meridian, and at what hour on a given day it is attained. 262. Determine the positions of all stars such that, when the aberration either in right ascension or declination vanishes, the other shall be a maximum. 263. Investigate a method of determining the longitude, by observing the distance of the moon from the sun or a fixed star. Why is the use of this method particularly convenient at sea? 264. Explain the method of determining the place of the node of a planet's orbit from observations made on the planet near its node. 265. Construct a vertical south-east dial. What is the greatest inclination of a vertical dial to the meridian, that it may shew the time of sunrise throughout the year? 266. Find when that part of the equation of time which is caused by the obliquity is additive or subtractive. Shew how to correct a watch by a sun-dial. SECTION XVII. QUESTIONS IN CENTRAL FORCES AND PHYSICAL ASTRONOMY. 1821 1. The moon revolves in a circle about the earth, and the quantity of matter in the earth is suddenly doubled. Compare the eccentricity of the orbit now described with its axis-major, and with the original radius of the moon's orbit. 2. Find the force of the sun to disturb the motions of the moon. 3. A pendulum vibrating seconds at the earth's surface is carried to a distance from the centre of the earth equal to that of the moon. What is the time of its vibration? 4. Find the time in which the moon would fall to the earth, if suddenly deprived of its angular motion. 5. In the 11th section of Newton, the mean motion of the nodes or apsides varies as the periodic time of P directly, and as the square of the periodic time of T inversely. 6. Find an expression for that part of the force of Jupiter to disturb Saturn, which acts in the direction of a tangent to Saturn's orbit. 7. If the accelerating gravity of a satellite of Jupiter to the sun was greater than that of Jupiter at the same distance, in the ratio of e: 1, where e differs little from unity, then the distance of the centres of the sun and of the orbit of the satellite, would be greater than the distance of the centres of the sun and Jupiter in the ratio of e: 1. 8. If the sun and moon be both supposed to be in the equator; to compare the lengths of a lunar and a tide day, for a given position of the luminaries. 9. In elliptical orbits of small eccentricity, the diminution of angular velocity in moving from the lower apse to the higher, is nearly proportional to the increase of distance. 10. A body describes an ellipse, the force being in the centre; given the force at a given distance, to find the actual periodic time. 11. A body is projected from a given point, in a given direction with a given velocity, when the force varies inversely as (dist.)2; find the latus-rectum of the orbit described. 12. The earth being a sphere, and its radius 4000 miles, what must be its diurnal rotation that a body at the equator may lose half its weight? 13. A body describes a logarithmic spiral, and approaches the centre by a space which is small compared with the whole distance. Compare the time of one revolution with the time to the centre. 14. At what distance from its centre must the earth, considered as a sphere, receive a single impulse, so as to produce its diurnal and annual rotation? 15. A comet describes 90° from the perihelion in 100 days. Compare its perihelion distance with the radius of a planet's circular orbit which revolves about the sun in 942 days. 16. If a body is projected in any direction, and acted upon continually by two forces tending to fixed centres, not both in the same plane with the direction of projection, it will describe by lines drawn from the two fixed points equal solids in equal times about the line joining the two points. 17. Find the mean horary motion of the moon's nodes, when 1822 the line of the nodes is in octants. 18. A body is projected in a given direction with a given velocity from a given point, and is acted upon by a repulsive force which varies as the distance from another given point. Required the curve which it will describe. 19. A body describes a circle about a fixed point, the force varying inversely as the square of the distance; another body, the attractive force of which varies in the same law, is introduced into the system; how will this affect the velocity of the body, the form of the orbit, and the periodic time? 20. Given the velocity of projection equal to the velocity in a circle at the same distance, (force a 1 D2); required the di rection in which a body must be projected at a given distance, that the focus of the conic section described may bisect the semi-axis major, and determine the magnitudes and positions of the axes. 21. Investigate the apsidal equation, and shew what number of possible positive roots it can have, when the velocity is acquired from a finite distance and the force varies as D”-1. 22. Find the horary motion of the moon's nodes in a circular orbit. (Newton, Book III. Prop. 30.) 23. The sum of the areas described by any number of bodies round a given point, multiplied by the respective masses of the bodies, is proportional to the time, if they be supposed to be acted upon only by their mutual attractions, and by the force tending to the given point. 24. Given the time, construct for the inclination of the lunar orbit to the plane of the ecliptic. (Newton, Book III. Prop. 35.) 25. Determine the point in P's orbit, (Sect. 11,) where the tangential ablatitious force is a mean proportional between the addititious and central ablatitious forces. 26. The velocity in an ellipse at the greatest distance is half that with which a body would move in a parabola at the same distance. What is the eccentricity of the ellipse? 27. Shew that the inclination of the moon's orbit is the greatest, when the line of the nodes is in syzygy; and the least, when the nodes are in quadrature and the moon in syzygy. 28. Two material points S and P, the mass of the first being twice that of the second, attract each other with a force which varies inversely as the square of the distance. When they have approached each other by half their original distance, P receives a new perpendicular impulse, which communicates to it a velocity equal to that which S has acquired. What curve is now described by each about the other? 29. Find the whole variation in the inclination of the moon's orbit, as the moon moves from quadrature to syzygy; the line of the nodes lying in quadrature. (Newton, Book III. Prop. 34, Cor. 4.) 30. Find the horary variation of the inclination of the lunar 1823 orbit to the plane of the ecliptic. (Newton, Book III. Prop. 34.) 31. Find the angular distance of a body from the vertex of a common parabola where the velocity is equal to half the greatest velocity. 32. The difference of the forces on P and p (Newton, apse with the velocity acquired in descending from an infinite distance to that point, construct the curve described, and find the time of descent to the centre. 34. Prove that the periodic time of a body revolving in an ellipse round the focus = Σπα = where a semi-axis, and m = force at a distance 1; and apply this result to deduce the actual time of falling down AC, in Prop. 32, Sect. 7. 35. Shew that in consequence of the mean disturbing force of the sun in the direction of the radius vector, the distance of the moon from the earth is increased by a 358th part, and her angular velocity diminished by a 179th part. |