A ray, passing through a point Q, is incident upon a refracting plate; q is the intersection of the emergent ray, produced backwards, with the normal to the plate through Q: if the angle of incidence be equal to tan-1μ, and t be the thickness of the plate, prove that

8. A ray of light passes through a prism in a plane perpendicular to its edge: shew that, if and y be the angles of incidence and emergence and i the refracting angle of the prism, the deviation is equal to

pi, or y − 4 – i,

according as the incident ray makes an acute angle with the face of the prism towards the thicker end or the edge. Under what convention will these expressions for the deviation be all represented by +-i, and with this convention for what value of will change sign?

9. Explain the formation of an image by reflection, and find the magnitude and position of the image of a given object placed before a plane mirror.

The faces of two walls of a room, meeting at right angles, are covered with plane mirrors: shew that a person will be able to see but one complete image of himself in either wall.

10. A diverging pencil of rays is incident directly upon a concave spherical refractor: find the geometrical focus of the refracted pencil.

A short object is placed perpendicularly on the axis of the refractor, and at a distance from it equal to ,f being the focal length: prove that the

f

μ

linear magnitude of the virtual image is half that of the object.

11. Describe the human eye as an optical instrument. When a pencil of rays is refracted through the eye, at what point of its passage does it experience its principal modification of form; and what is the most probable hypothesis in regard to the change of configuration of the eye by which it adjusts itself to distinct vision at different distances?

An eye is placed close to a sphere of glass, a portion of the surface of which, most remote from the eye, is silvered: prove that, assuming eight inches to be the least distance of distinct vision, the eye cannot see a distinct image of itself unless the diameter of the sphere be at least ten inches in length.

12. Describe Galileo's telescope, and trace a pencil of rays through it. State what would be the effect on the image

(1) of increasing the size of the object-glass

สล

(2)

} eye-glass unaltered;

THURSDAY, January 8, 1857. 9...12.

1. ENUNCIATE and prove Newton's fourth Lemma.

Apply this Lemma to shew that the volume of a right cone is one-third of that of the cylinder on the same base and of the same altitude.

2. Enunciate Lemma XI, and prove it when the subtenses are parallel.

An arc of continuous curvature PQQ', is bisected in Q; PT is the tangent at P; shew that ultimately, as Q'approaches P, the angle Q'PT is bisected by QP.

3. Shew that, if a subtense be drawn from the extremity of an arc of finite curvature, in any direction, the chord of curvature parallel to that direction is the limit of the third proportional to the subtense and the arc.

Hence find the chord of curvature through the focus at any point of an ellipse; and prove that half this chord is a harmonic mean between the focal distances of the point.

4. State and prove Proposition 1.

Will the velocity of the body or the rate at which areas are swept out about the centre of force be affected by any sudden change in the law of force?

A body moves in a parabola about a centre of force in the vertex ; shew that the time of moving from any point to the vertex varies as the cube of the distance of the point from the axis of the parabola.

5. A body is revolving in an ellipse, find the law of centripetal force tending to the centre of the ellipse.

Shew that the time in which any given area will be swept out by the radius vector is independent of the eccentricity of the ellipse, if the area of the ellipse be given.

6. If any number of bodies revolve in ellipses about a common centre, and the centripetal force varies inversely as the square of the distance; the squares of the periodic times are proportional to the cubes of the major

axes.

A particle moves in an ellipse about the centre of force in the focus S: when the particle is at B, the extremity of the minor axis, the centre of force is changed to S' in SB, so that S'B is one-fifth of SB, and the absolute force is diminished to one-eighth of its original value; shew that the periodic time is unaltered, and that the new minor axis is two-fifths of the old.

7. Define the term "zenith," and explain some method for determining the zenith of a given observatory.

How would an increase in the Earth's velocity of rotation affect the latitude of a given place, supposing the form of the Earth to remain unaltered ?

8. What conditions must be satisfied in order that the transit instrument may be in accurate adjustment?

Shew how by aid of this instrument the difference in right ascension of two stars may be determined; and state the principal astronomical assumptions on which the truth of this determination depends.

9. Explain the phrases "mean solar time" and "equation of time". Shew that in the month of February the equation of time is additive. Account for the fact that the time of the Sun's setting as given in the ordinary Almanacs is not the latest on the longest day?

10. Prove that generally the apparent place of a star will depend upon the ratio of the velocity of the Earth in her orbit to the velocity of light.

Find the least diurnal velocity of rotation of the Earth, which will render sensible to an observer at the equator the aberration due to this cause, the least appreciable angle being '1".

11. Describe the apparent motion of the Moon among the stars, and the real motion of its centre of gravity about the Sun, illustrating the latter description by a figure.

What is inferred from the fact that, with slight variations, the same portion of the Moon's surface is always presented to the Earth? How much should the Moon's rate of rotation about its centre of gravity be increased, in order that its whole surface might be seen in the course of one orbital revolution?

12. Explain the method of determining the longitude by means of Lunar Distances.

On January 1st, 1855, at the mean time 9 hrs. 42 min. 8 secs. P.M., the distance of a Arietis from the Moon's centre was calculated from observations to be 45° 30′ 16′′: at noon and at 3 P.M. Greenwich mean time, the distances are 44° 56′ 11′′, and 46° 23′ 39′′ respectively: find the longitude of the place of observation.

THURSDAY, January 8, 1857. 1...4.

1. THREE circles, A, B, C, intersect in a common point, the other intersections of (B, C), ( C, A),(A, B), being a, ß, y, respectively. If b, c, be points in B, C, respectively, such that b, a, c, lie in a straight line, prove that a, the intersection of by, cß, produced, lies in the circle A.

2. Shew that the sum of all the harmonic means, which can be inserted between all the pairs of numbers the sum of which is n, is

4. From a point on a hill-side of constant inclination the angle of elevation of the top of an obelisk on its summit is observed to be a, and, a feet nearer to the top of the hill, to be ß; shew that, if h be the height of the obelisk, the inclination of the hill to the horizon will be

5. Each of three circles, within the area of a triangle, touches the other two, touching also two sides of the triangle: if a be the distance between the points of contact of one of the sides, and b, c, be like distances on the other two sides, prove that the area of the triangle, of which the centres of the circles are the angular points, is equal to

} (b3 c2 + c2 a2 + a2 b3)‡.

6. The acute angles, which the distances of two points of an ellipse from the same focus make with the respective tangents at the points, are complementary to each other: prove that the square on the semi-axis minor is a mean proportional between the areas of the two triangles, of which the two points are the respective vertices, and the distance between the foci the common base.

Shew that the problem is impossible unless the axis minor is less than the distance between the foci.

7. CP, CD, are two conjugate semi-diameters of an ellipse: Tt is a tangent parallel to PD: a straight line CIJ cuts at a given angle PD, Tt, in I, J, respectively: prove that the loci of I, J, are similar curves.

8. A fine string ACBP, tied to the end A of a uniform rod AB of weight W, passes through a fixed ring at C, and also through a ring at the end B of the rod, the free end of the string supporting a weight P: if the system be in equilibrium, prove that

AC BC: 2P+W: W.

9. A picture is hung up against a rough vertical wall by a string fastened to a point in its back, so that the picture inclines forwards; apply the principle of the triangle of forces to find the inclination of the string to the wall, when its tension is the least possible.

10. A lamina, cut into the form of an equilateral triangle, is hung up against a smooth vertical wall by means of a string attached to the middle point of one side, so as to have a corner in contact with the wall; shew that, when there is equilibrium, the reaction of the wall and the tension of the string are independent of the length of the string, and that, if the string be beyond a certain length, equilibrium in such a position is impossible.

11. A ball is projected from the middle point of one side of a billiard table, so as to strike in succession one of the sides adjacent to it, the side opposite to it, and a ball placed in the centre of the table; shew that, if a and be the lengths of the sides of the table and e the elasticity of the ball, the inclination of the direction of projection to the side a of the table from which it is projected must be

12. A perfectly elastic ball is projected at an inclination ẞ to a plane inclined to the horizon at an angle a, so as to ascend it by bounds; find the inclination to the plane at which the ball rises at the nth rebound, and shew that it will rise vertically if

cot ẞ= (2n+1)tan a.

13. A string, charged with n+m+1 equal weights fixed at equal intervals along it, and which would rest on a smooth inclined plane, with m of the weights hanging over the top, is placed on the plane with the (m+1)th weight just over the top; shew that, if a be the distance between each two adjacent weights, the velocity which the string will have acquired, at the instant the last weight slips off the plane, will be

{nag}*.

14. A perfectly elastic ball is projected with a given velocity from a point between two parallel walls, and returns to the point of projection, after being once reflected at each wall; prove that its angle of projection is either of two complementary angles.