4. Having given the ratio of the sines of incidence and refraction, when a ray passes out of one medium into each of two others, to find the ratio of the sine of incidence to the sine of refraction out of one of the latter mediums into the other.

5. When parallel rays are incident nearly perpendicularly upon a spherical refracting surface, the distance of the intersection of the refracted ray and the axis, from the centre, is the greatest when the arc is evanescent.

6. A pencil of parallel rays passes from water through a globule of air ; find the focus after the second refraction.

7. Find the focal length of a compound lens.

8. Explain the construction of Newton's telescope, and shew how it must be adjusted to the eye of a long-sighted person.

9. An object whose' real depth below the surface of the water is ten feet, is viewed by an eye fifteen feet above the surface. What must be the focal length of a lens through which it is viewed, that its apparent depth may be ten feet ?

10. If a plane mirror be turned round uniformly, the angular velocity of the image of a given object formed by continual reflection at its surface : angular velocity of reflector :: 2:1.

11. The radii of a spherical reflector and sphere of glass of same aperture and power are in the proportion of 3:1. Compare the density of rays in sun's image formed by them.

12. Two straight lines are inclined at a given angle, and a point E is given without them, a line E F f moves round the point E, and cuts the given lines in F and f; find the locus of the mirror D, so that f shall always be the image of F.

13. Suppose a mirror M to move in a straight line AB, and an object D in the line AC at right angles to it, and distance between object and mirror to be constant; to determine the locus of an eye, which being always at the same distance as the object from the mirror, shall always see the object.

14. Having given the refracting powers of two mediums, to find the ratio of the focal lengths of a convex and concave lens, formed of these substances, which, when united, produce images nearly free from colour.

15. When a ray of light is incident obliquely upon a spherical reflector, to determine the intersection of the reflected ray and the axis of the pencil to which it belongs, and shew that when the focal length is given, the longitudinal aberration of parallel rays varies as (lin. apert.)2 and lateral aberration varies as (lin. apert.)s.

16. If an objecť be placed in the principal focus of a double convex lens, the visual angle is the same, whatever be the distance of the eye from the glass.

17. Find the length of a caustic generally, and apply it to the case when the reflecting curve is a semicycloid, rays parallel to axis.

18. Find the density of rays in a caustic, when reflecting surface is a hemisphere, radiating point in surface.




1. What are the situations of the Sun, Moon, and Earth at the time of full Moon—of new Moon—of half Moon? What are the situations of the bodies at the time of a Lunar eclipse ?

2. The longest day is (n) hours more than a day at the equator: what is the latitude of the place ?

3. Given the synodic year of a Planet ; compare the radius of its orbit with that of the Earth's.

4. What is the elongation of Venus from the Earth when she will appear stationary? Their mean daily motions being known, compute the time when that appearance will take place.

5. Find the time of shortest twilight for a given north latitude: and shew that then the Sun's azimuths at the beginuing and end of twilight are supplements of each other, and that the Sun's declination is south.

6. A Star, as it leaves the prime vertical, is observed to have an increase of azimuth, which is, to the contemporary increase of alti. tude, :: a : b. Required the latitude of the place.

7. The latitudes of two Stars which have the same right-ascension, are given, and also their difference of longitude. Deduce the longitude of each and the right-ascension.

8. A telescope deviates from the plane of the meridian by a small angle: and it is observed, that the time elapsed between the appulses of two known Stars to it is not equal to their difference of right-ascension converted into time. Correct the deviation.

9. The increments of gravity in moving from the equator to the pole, are proportional to :he sin. of lat.; the Earth being supa posed a sphere.

10. Given the lengths of two pendulums vibrating seconds in two places of known latitude. Find the eccentricity of the ellipse, by the revolution of which round the polar diameter, the Earth would be generated.

il. The Earth moves in an ellipse round the Sun in the focus : Required the law of the force by which it is retained in the orbit.

12. The periodic times in different ellipses round the same centre of force, are in the sesquiplicate ratio of the major axes.

13. Prove the law of variation of the angular velocities of different bodies, revolving in their orbits round the same centre of force ; and find when that part of the equation of time arising from the unequal angular motion of the Sun in the ecliptic is a maximum.

14. Find the true anomaly in terms of the eccentric.

15. Give a formula for clearing the Moon's distance of the effects of refraction and parallax.

16. Correct the meridian line as determined by the method of equal altitudes—the Sun having changed his declination by a given quantity, during the interval of the two observations.

17. What would be the curve of aberration of a Star, if the Earth were to move in an ellipse round the centre?

18. Find the effect of aberration on the right-ascensions and declinations of Stars. Investigate, in each case, the position of the Earth when that effect = 0.

19. Investigate the dimensions of the Conic section traced out in the course of a given day at a given latitude, by the extremity of the shadow cast upon an horizontal plane, by a style perpendicular to it.

20. Explain the principle of the Sextant: and the division and use of the Vernier.

21. Compare the densities of the Sun and Earth; the periodic times of the Earth and Moon, and the Moon's horizontal parallax being known.




1. By what arguments is it inferred that the Earth revolves about its axis, and about the Sun ?

2. Find the right ascension and declination of a comet, when its distance from two known stars is given.

3. Required the time of the Sun's transit over the vertical wires of a telescope, on a given day at a given place.

4. The apparent meridian altitude of the Sun's lower limb = 53° 40', his apparent semi-diameter = 15° 50', his mean refraction=29", the parallax=4”.5, collimation = 34".5, and declination = 16° 13'. Find the latitude of the place of observation.

5. The altitude of the Sun was observed to be half of his declination at 6 o'clock. Prove that twice the sine of the latitude of the place the secant of the Sun's altitude.

6. There are two places on the same meridian, whose latitudes VOL. II.

2 Y


are the complements of each other; and on a given day the Su rises an hour sooner at one place than at the other. Required the latitudes of the two places.

7. If A and a be the altitudes of a star, on the same vertica! circle on the same day, (d) the declination of the star, and (1) the

Aca sin. I

2 latitude of the place,

; Required a proof. sin.d

Ata sin.

2 8. Construct a vertical south-east dial, for a given place. 9. When the Sun is in the equinoctial, the locus of the extremities of the shadow cast by a perpendicular object upon an horizontal plane is a straight line.

10. If a body revolve uniformly in a circular orbit, it is retained in that orbit by a force which tends to the centre of the circle ;

1 and if the periodic times in such circle o R", the force o

11. Determine the quantity of refraction by observations made upon the circumpolar stars. When will this method fail?

12. The parallax of a planet in right ascension being given, it is required to find the distance of the planet from the earth's centre, the earth being supposed spherical.

13. The sine of the excentric anomaly: the sine of the true anomaly :: the radius rector : the semi-axis minor.

14. Find the distance of a planet from the Sun.

13. Suppose an eclipse of the Moon to last three hours; to how mata prtion of the Earch will some part of it be visible?

10. Frore tè at when the first point of Aries rises, the ecliptic makes the least ange with the horizoa; ad explain from thence the phenomena of the hatest Mocn.

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1. If two sides of a spherical triangle be equal to a semicircle, the arc drawn from the vertex bisecting the base is a quadrant.

2. If two sides of a spherical triangle be less than a semicircle, the angle opposite the less side is less than 90 degrees.

3. If a great circle be 'stereographically projected, the radius of the projection is the secant, and the distance of its centre from the centre of the primitive is the tangent of the inclination of the circle to the primitive.

4. Having given the Sun's declination and diameter, and the latitude of the place; find how long the disk is in rising.

5. Find the absolute quantity of refraction for considerable altitudes according to Boscovich's method.

6. On a certain island the Sun was observed to be vertical when on the meridian. The declination was found from the tables to be D degrees ; and the time of sunrise at Greenwich to be T, and of sunset t hours before that observed by the chronometer. Find the distance and bearing of the place from Greenwich.

7. Having given the Sun's rising amplitude and altitude when on the prime vertical, determine the latitude of the place.

8. The Sun's declination which is given, is greater than the co-latitude which is also given. Determine the nature and dimensions of the curve, which is traced out by the end of the shadow of a vertical stick on an horizontal plane.

9. Determine, from three observations of a spot on the Sun's disk, the inclination of the Sun's equator to the ecliptic, the longitude of the ascending node, and the time of the Sun's revolution on its axis.

10. When a planet is stationary, the horary change in elongation is to the horary change in annual parallax as the periodic time of the planet to the periodic time of the Earth.

11. Find the shape and dimensions of the Moon's penumbra. 12. Explain how the Sun's parallax may be found by the transit of the inferior planets.

13. By the effect of aberration, a Star's latitude is least when the Sun's place is 90 degrees before the Star's place in the ecliptic, and from that time the increment in latitude varies as the versed sine of the Sun's longitude, reckoned from that point. 14. The longitude is least when the Sun is in syzygy with the

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