ST. JOHN'S COLLEGE, JUNE 1829.

1. WHEN diverging or converging rays are incident nearly perpendicularly upon a spherical reflector, the conjugate foci lie on the same side of the principal focus.

2. The concavity of a meniscus of glass of inconsiderable thickness is filled with water; the radii of the surfaces are 5 and 6 inches required the focal length of the compound lens.

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3. The surfaces of a concavo-convex lens of inconsiderable thickness are formed by the revolution of an ellipse round its major and minor axes. Given the focal length of the lens and the axes of the ellipse, determine the refracting power of the medium.

4. In the astronomical telescope, the linear magnitudes of the visible area and the bright part of the visible area are

where D, F and d, ƒ are the diameters and focal lengths of the object and eye glasses.

5. A ray of light parallel to the axis of a paraboloid of glass after refraction cuts the axis in a given point; required the point of incidence.

6. A fire is placed in the axis of a concave mirror, which reflects one-eighth part of the heat, at a greater distance from it than the principal focus. Where must a small object be placed in the axis between the fire and its geometrical focus that it may be the least heated?

7. Two lights are placed in the opposite extremities of the diameter of a circle, the intensities of which are :: 4:3. What part of the curve is the most illuminated?

8. What spherical reflectors and lenses from aberration are more powerful in the centre than at a distance from it, and the converse; and how are the effects of this imperfection partially counteracted in the telescopes of Galileo and Cassegrain?

9. If the cosines of incidence and refraction be to one anotherm 1, the surface, which will refract parallel rays accurately

to a point S, is formed by the revolution of the curve r = a

round an axis drawn through S in the direction of the incident rays; where r is the distance of any point from S, O the inclination of r to the axis, and a an arbitrary value of r when 0 = 90o.

10. A luminous body is surrounded by a medium which absorbs light with a power varying inversely as the distance from it. At the distances 1 and the intensities of light are " and "; prove that at the distance r the intensity is " . pn—m.

11. Prove that the altitude of the primary rainbow-4R-21-S;

for red rays. What is the corresponding formula for the secondary rainbow?

12. When a small pencil of homogeneal rays falls obliquely upon a spherical refractor, in a plane which passes through its centre, having given the focus of incident rays and the angles of incidence and refraction, required a formula to determine the geometrical focus of refracted rays.

ST. JOHN'S COLLEGE, MAY 1830.

1. COMPARE the illuminations of a horizontal and an inclined plane, the sky being covered with clouds of uniform brightness.

2. Having given the breadth of a man's face, and the distance between his eyes: find the breadth of the narrowest plane mirror in which he can see the whole of his face.

3. Find the successive images of a point placed between two concentric concave reflectors, of equal focal length.

4. Find the nature of the caustic, when the reflecting surface is a spheroid, and the rays are incident parallel to its axis.

5. Compare the real and apparent diameters of the bore of a barometer tube.

6. When a ray of light passes in any direction through a prism, the incident and emergent rays make equal angles with the edge of the prism.

7. Find the angle subtended by a small object viewed through a lens not placed close to the eye.

8. In the above question find the distance of the object from the lens, the focal length of the lens being F, its distance from the eye k, and the distance of most distinct vision c.

9. Find the radii of the surfaces of a lens of given focal length, when the longitudinal aberration for parallel rays is a minimum, the value of being 1.5.

10. When a pencil of homogeneal rays is refracted into a sphere in such a manner that the rays emerge parallel after two internal reflections, determine the ultimate intersections of the rays in the plane of incidence. Determine also their ultimate intersection (after the first refraction) in a plane perpendicular to the plane of incidence.

11. When a beam of white light is incident on a prism in a plane perpendicular to its edge, the angle through which the emergent pencil is dispersed,

sin.I. secant, secant .d nearly.

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12. Describe the Magic Lantern. If instead of having one lens between the picture and screen, it has two of equal focal lengths 3c placed at the distance 2c from each other, find the least distance between the picture and screen.

13. The focal lengths of the object-glass, field-glass, and eye-glass of a telescope, are 60c, 6c, and 2c respectively, the diameter of the object-glass is 3c, that of the eye-glass is c: find the power of the telescope, and compute roughly the diameter of the field-glass when just large enough not to contract the field of view.

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QUEEN'S COLLEGE, 1824.

1. A GIVEN vertical object is supposed to be a given height above an horizontal plane, required the point in the plane from which the object will appear to have the greatest magnitude.

2. Two points and an indefinite straight line placed between them, not at right angles to a line joining the points, being given in position; find a portion of the line such, that its apparent magnitude seen from one of the points may be double of the apparent magnitude seen from the other.

3. If a plane mirror revolve about an axis, the angular motion of the image of any object is double of the angular motion of the mirror.

4. A radiant point and the position of the eye being given, both on the same side of a plane mirror; if the mirror be supposed to move in a direction perpendicular to its own plane, required the locus of the several points of the mirror from which rays are reflected to the eye.

5. It is required to find an incident ray parallel to the axis of a given reflecting circular arc, which shall be reflected so as to pass through a given point in the axis.

6. If a ray parallel to the axis of a glass sphere, one half of which is covered with a reflecting substance, fall upon the refracting half of the sphere; after refraction it shall be reflected to a point in the axis, at a distance from the reflecting surface nearly equal to one sixth part of the sphere's diameter.

7. Having given the distance of an image from a double concave lens, and the ratio of the object to the image; required its focal length.

8. How many radii of the convex surface of a plane convex lens must an object be placed from it, so that its magnitude may be (m) times greater than that of its inverted image.

9. An object viewed with a Gregorian telescope is magnified (m) times the focal length of the eye-glass is (a) inches, and the distance between their principal foci given; required the relation of the focal lengths of the two reflectors.

10. If the thickness of two microscopic glasses be of their radii of convexity, and these be in the ratio of 10 to 3, how must the glasses be disposed in a double microscope, so that an object 8 inches distant from the eye may be magnified 1000 times.

11. When parallel rays are incident upon a spherical reflector, show that the radius of the least circle of aberration varies directly as the cube of the semi-aperture, and inversely, as the square of the focal length of the reflector.

12. If the Sun be on the horizon, what is the altitude of an hill from the top of which a rainbow will appear to be perfectly circular?

13. The radiating point and the caustic being given, show that there are an infinite number of reflecting curves which will produce the caustic.

QUEEN'S COLLEGE, 1827.

1. PLACE an object before a double convex lens, so that the image may be double of the object, and erect.

2. Determine the apparent magnitude of a straight line, placed at a given depth, parallel to the surface of a vessel of water, the eye being situated in a given point in the plane passing through the object perpendicular to the surface.

3. Two plane reflectors are inclined to each other at an angle of 30° required the number of images of an object situated between them.

4. When the reflecting curve is a cycloid, and the rays proceed parallel to the axis, find the caustic.

5. Find the magnifying power of a simple microscope.

6. Explain the Camera Lucida.

7. In a double convex lens, find the position of the conjugate foci when their distance from each other is a minimum.