2. A ray of light cannot pass out of a denser into a rarer medium, when the angle of incidence exceeds a certain limit when sin. inc. sin. ref.:: 1: 2,

3. S and H are fixed points; a ray SP proceeding from S, is reflected to H by a plane mirror AP moveable above a given point A in the line SH produced; required the curve, which is the locus of P.

4. Given the distance (a), to which a short-sighted person can see distinctly; find at what distance from his eye a concave lens of given focal length (v), must be placed, to enable him to see an object Q at a distance (b); and explain what is to be understood by that result, which places the lens beyond Q.

5. The whole circumference of a given circle is luminous, except a very small arc, which reflects the light from the vertical. Required the form of the caustic.

:

6. Sin. inc. sin. ref. :: m : 1; rays diverging from a given point Q, enter nearly perpendicularly into a sphere, the opposite surface of which is quick-silvered. Find the geometrical focus of emergent

rays.

7. The image of a straight line formed by a single spherical refracting surface is a conic section; shew that if the line be placed within a sphere of glass, the image must be an hyperbola. And explain how we may suppose it to be completed.

8. If Q be the focus of rays incident on a spherical refracting surface, and Q' the focus of incident rays coming in the contrary direction, such that Q and Q' have the same conjugate focus, required a maximum and minimum value of Q Q'.

9. There is a small aperture of the shape of an isosceles double convex lens of inconsiderable thickness, in the centre of a glass sphere; if this aperture be filled with a given substance, find the focal length of the sphere in air. Supposing m and m′ to be the sin. inc. values of out of air into glass, and the interior substance sin. ref.

respectively.

10. If parallel rays be incident in a curve in a direction perpendicular to the axis of a; prove that the co-ordinates x', y', to the point where the reflected ray meets the caustic, are determined by the

and hence shew that the caustic to the logarithmic curve

is the common catenary.

11. Find the longitudinal aberration when a pencil of parallel rays is incident on a spherical refracting surface; and if (0) be the

angle at which the extreme ray is incident, and

sin. inc.
sin. ref.

=m, shew

Ө

that the lateral aberration = rad. sin. vers.

nearly.

-

m

12. Construct Cassegrain's telescope, and find its field of view, also compare its magnifying power with that of a Newtonian telescope, having the same eye-glass, and large reflector.

ST. JOHN'S COLLEGE, MAY 1826.

1. EXPLAIN the construction of the single microscope, and find its magnifying power.

2. At what point in the axis of an elliptical reflector must Q be placed that q may be at the extremity of the axis major.

3. A glass sphere is divided into 3 parts by two parallel planes which trisect its diameter at right angles. The glass between the planes is removed, and its place supplied by water. Find the distance between the images of two objects placed in the centre, and at the extremity of the diameter.

4. Q is an object placed between two inclined plane reflectors. Find the locus of an eye such that the path of the ray (from Q to the eye) by which the nth image of one side of Q is seen may be of the same length as that by which the nth image of the other side of Q is seen.

5. Of how many degrees must a polished circular arc be, that the image of its tangent may be a quadrant of an ellipse?

6. A straight line is inclined at a given angle to the side of a glass prism whose vertical angle is known. Find the nature and position of the image.

7. If a Gregorian telescope be adjusted to the eye of a shortsighted person, and the eye be farther from the lens than the principal focus, what change is produced in the visual angle, and in the position of the image on the retina?

8. Find the thickness of a plane glass mirror that the distance of the image from the first surface may be twice as great as in a mirror of inconsiderable thickness.

9. Investigate the nature of the reflector that parallel rays incident upon it in the same plane may be reflected parallel.

10. When diverging rays are refracted by a medium contained by parallel plane surfaces, the aberration is less than when they are refracted by a single surface.

11. Find the form of the surface of a medium such that rays proceeding from a point without it may be refracted accurately to a given point within it.

ST. JOHN'S COLLEGE, MAY 1827.

1. IF a man 6 feet high be placed before a vertical mirror nearly his own height, at a distance of 4 feet, required how much of his person will be illuminated by a lamp behind him, 12 feet from the mirror and 11 feet from the ground; and find his position when his feet are just illuminated.

2. A ray of light suffers as much deviation in passing through one medium contained by parallel plane surface into another, as in passing immediately into the latter medium.

3. Given the focal length of a lens, which will just enable a short-sighted person to receive parallel rays, to determine the focal length of one, which will enable him to see to any given distance.

4. Rays tending to form an image at the point where the eye is placed, are received upon a concave lens, required to prove that the visual angle varies inversely as the square of the distance of the lens from the eye.

5. Given the focal length of a glass lens in air, find another, which, being compound with it, and the whole placed in water, the focal length shall equal that of the first in air.

6. When rays are incident parallel to the axis of a cycloid, to determine the form of the caustic and the law of the density.

7. Rays diverge from a point in the axis of double convex lens whose thickness equals one of its radii. Required the geometrical focus of refracted rays.

8. The image of a straight line perpendicular to the axis of a concave spherical reflector, and passing through its centre as seen by an eye placed in the axis at such a distance that all lines drawn to it from the reflector may be considered parallel, is determined by the equation

y.(r2 + 2x2)= 2x2 / (1o — x2).

Investigate this, and trace the curve.

9. In the last question, find in what part of the reflector the image of the diameter is formed. And explain clearly where the eye must be supposed to be placed when the image of a straight line formed by a spherical reflector is said to be a conic section.

10. An homogeneous pencil of parallel rays is incident on a sphere of refracting substance, so as to emerge parallel after (p) internal reflections; required the angle between the incident and emergent pencils.

11. The lunar apertures of the object and eye-glass of an astronomical telescope being as 6: 1, and their focal lengths 27 inches, and inches respectively, required the focal length of a convex lens to be placed between them one inch from the eye-glass, and of twice its aperture, that the field of view may be doubled, when adapted to common eyes. Also determine the effect upon the magnifying power of the telescope.

12. Compare the focal lengths of two lenses of different given substances, which when compounded shall cause the red and violet rays of a pencil to converge to the same focus; and shew that if the spectrum formed by similar prisms of each substance were divided exactly in the same proportion by the different colours, the compound lens would be wholly free from colour.

ST. JOHN'S COLLEGE, MAY 1828.

1. EXPLAIN by what contrivances in the eye objects are seen distinctly at different distances and with different degrees of light. 2. Given Q the focus of rays incident nearly perpendicularly on a concave reflecting sphere; find q the focus after two reflections. 3. Find by experiment the principal focus of a lens, or spherical reflector.

4. Archimedes' burning mirror is supposed to have been formed by small planes reflecting the Sun's rays to the same point. Find the surface they all touch, and explain what limits its effects.

5. Given the interior, find the exterior radius of a meniscus by which a person who can see distinctly at one given distance, may be enabled to see distinctly at another, and explain the advantage of this form of lens for spectacles.

6. If sin. I : sin.R;: √3 : 2, out of the air into a sphere, find what part of the Sun's rays incident on the sphere are transmitted. 7. Given Q the focus of rays incident nearly perpendicularly on a prism AIC, find q the focus of refracted rays, and also the curve described by q if the prism increase in thickness by the revolution of the side IC round I.

8. Compare the magnifying powers of a sphere and a planoconvex lens of the same radius, and shew by what contrivance the two may be made equal, and the advantages of a plano-convex lens so formed.

9. Given the diameters and focal lengths of the object and eyeglass in the Astronomical telescope; find the area of the bright part of the field of view, and explain the use of the stops and additional glasses usually found in telescopes.

10. In Cassegrain's telescope the spherical aberrations partly correct each other, but cannot be made to do so entirely.

11. If the dispersive powers of two media are as 3; 2, find the focal lengths of two lenses which will, combined, produce an achromatic lens of 12 inches focus.

12. Find the caustic of rays diverging from a luminous point, and reflected by a plane silvered mirror.