8. Find the centre of pressure of a semi-parabola immersed vertically in a fluid, with its vertex just coincident with the surface and its base parallel to it.

9. A vessel in the form of the hyperboloid of revolution has its axis vertical and vertex downwards; find the time of emptying it through a given small orifice in the vertex.

10. A given paraboloid contains a given quantity of fluid; determine the angle at which its axis may be inclined to the vertical before the fluid runs out.

11. A fluid mass has a given uniform motion of rotation, and its particles are attracted to two constant centres of force in the axis of rotation. Determine the form which its surface will assume.

12. A body oscillates in a circle, in the medium of which the resistance varies as the velocity; find the time of one of its small oscillations.

13. Find the lengths of the day and night, when

I being the latitude of the place and ♪ the Sun's declination.

14. The Sun's azimuths at the beginning and end of twilight are supplements to each other on the day of shortest twilight.

15. Prove that the times in seconds in which the Sun will pass the horizontal aud vertical wires of a telescope are respectively

D"

and

15 cos. lat. sin. dec.

D"
15 cos. dec.

D being the apparent diameter.

16. A style projects vertically from the top of an upright cone; find the path described by the extremity of its shadow on the surface of the cone.

OPTICS.

TRINITY COLLEGE, 1820.

1. GIVEN the distance of the focus of incident rays from the centre of a given spherical reflector; find the distance of the geometrical focus of reflected rays from the centre, when they are incident nearly perpendicularly.

2. Given the position of an object placed between two plane reflectors inclined at a given angle; find the total number of images, and apply it to the case where the angle of inclination is equal to

11° 15'.

3. A straight line is placed before a concave spherical reflector, at the distance of one-third of its radius from the surface; find the dimensions of the curve formed by its image, the radius of reflector being 9 inches.

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 EFf moves round the point E, and cuts the given lines in F and ƒ; 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.) and lateral aberration varies,as (lin. apert.).

16. If an object 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.

TRINITY COLLEGE, 1821.

1. (1). STATE the velocity of light, and the manner in which it was first ascertained.

2.

(2). Prove that the intensity of light, when transmitted through an uniformly dense medium, decreases in geometrical progression.

Enumerate the principal discoveries of Newton in the science of Optics; and give a concise account of the experiment by which he determined the refrangibility of light.

3. If an object be placed before a parallel plane mirror, the image will appear to occupy half the space occupied by the object.

4. Two fixed objects situated on the same side of an indefinite straight line, are viewed from different points of that line; required the point at which their apparent distance shall be a maximum.

5. Rays nearly parallel being incident on a spherical reflector whose radius is (r); prove, that if d, d' be the distances of the foci of incident and reflected rays from the surface

according as the reflector is concave or convex.

6. Find an incident ray parallel to the axis of a given reflecting circular arc, that shall be reflected so as to pass through a given point in the circumference.

7. The reflecting curve being the common parabola, and the incident rays perpendicular to the axis, find

(1). The caustic, and

(2). The point where it intersects the axis.

8. If a ray of light be incident obliquely on a small arc (0) of a spherical reflector, whose radius is (r), and if (d) be the distance of the focus of incidence from the centre; prove that the longitudinal aberration

9. An infinitely slender pencil of rays is incident nearly perpendicularly upon a spherical refracting surface whose radius is (r); then if (d), (x) denote the distances of the foci of incident and refracted rays from the surface, and if the sine of incidence be (n) times that of refraction,

10. (1), If an homogeneous ray of light be refracted through a prism of a denser medium, the deviation will be a minimum, when the refractions on both sides of the prism are equal.

(2). Shew that when this is the case, if (a) be the vertical angle of the prism, (d) the deviation, and (n) the constant ratio of the sine of incidence to that of refraction.

sin. (a)
sin.a

(3). By what methods

may the refractive power of any transparent solid substance be determined?

11. Required the focal lengths of a double convex lens of given thickness, the sine of incidence being (") times that of refraction.

12. Explain the structure of the eye, and the manner in which vision is performed.

13. Shew in what positions of the eye and object, the image formed by double convex lens will be

(1). Magnified or diminished: and

(2). Erect or inverted.

14. (1). Construct Gregory's telescope.

(2). Prove that the field of view depends more upon the aperture of the eye-glass, than of the smaller reflector: and

(3). Shew that the aberration produced by the first reflection, is increased by the second.

15. Given the curvature and aperture of a plano-convex lens, whose plane part is directed towards the object; compare the errors arising from spherical and chromatic aberration.

16. Find at what point in the axis of a given object-glass, a lens of known focal length must be placed, so as to correct the dispersion of an heterogeneous ray refracted through the former