of any substance, solid or fluid; and find the angle at which a ray must be incident upon a prism so that the deviation in passing through the prism may be a minimum, the whole course of the ray being in a plane perpendicular to the axis of the prism.

6. Find the geometrical focus of rays after refraction at a spherical surface of a refracting medium: and determine that focus of incidence which will have its geometrical focus of refracted rays at the least distance from it: also that focus of incidence for which there will be no aberration.

7. Find the principal focus of a sphere, and apply the expression to the case of a sphere of glass placed in a medium of water, the luminous point being within the medium.

8. Define the centre and focal centres of a lens, and determine the latter for the meniscus.

9. Parallel rays are incident upon a double convex lens of which the second surface is silvered; the radii of the lens are 5 and 7 inches respectively, and the index of refraction 2·45. Find the focus of emergent rays.

10. Determine the spherical aberration of a lens, and shew that it varies as the square of the radius of the aperture.

11. Explain fully the construction of the human eye, and shew to what optical instrument it is very nearly analogous.

12. Determine the diameter of the least circle of chromatic aberration.

13. Define dispersive power, and construct an achromatic prism.

14. Prove that rays which after reflection or refraction tend to form an image, may be refracted to the eye by a double concave lens, so as to form a distinct image on the retina: and determine the variation of the visual angle in this lens.

15. Construct and explain Cassegrain's telescope, find its magnifying power, and define the boundaries of the field of view: shew also the advantages of this telescope over that of Gregory.

16. When contiguous parallel rays fall upon a spherical drop of water and emerge parallel after any number of reflections, find the angle between the incident and emergent rays. Explain the forma

tion of the rainbow, and the cause of the inverted order of the colours in the primary and secondary bows, and determine the breadth of them.

17. Find the form of the surface of a medium which will refract rays diverging from a point without it accurately to a point within itself.

18. The caustic formed by rays after refraction at a plane refracting surface is the evolute to the ellipse or hyperbola, according as the medium is denser or rarer.

NEWTON.

TRINITY COLLEGE, 1820.

1. EXPLAIN by short examples, the method of exhaustions, of indivisibles, and of prime and ultimate ratios.

2. Prove that if a radius vector be drawn bisecting any arc, it must ultimately bisect the chord.

3. If a straight line EDA make with the curve CBA a given angle at the point A, and the ordinates CE, BD be drawn; the triangles ACE, ABD are ultimately in the duplicate ratio of the sides.

4. Let AB be the subtense of the arc, AD the tangent, BD the subtense of the angle of contact perpendicular to the tangent, as in the 11th Lemma: then let a series of curves be drawn in which DB x AD1, AD, AD, &c., the angle of contact in each succeeding case will be infinitely less than in the preceding.

5. If the areas described by the radius vector are not proportional to the times, the revolving body is not acted on solely by a force towards a fixed centre.

6. If a body be acted on by a given force and revolve in a circle, the arc described in any given time is a mean proportional between the diameter of the circle and the space through which a body would descend in the same time from rest if acted on by the same force.

7. The velocity at any point of a curve described round a centre of force = the velocity which a body, acted on by the given force at that point, would acquire by descending through part of the chord of curvature.

8. Given the force of gravity = 32 feet, and the radius of the earth = 4000 miles; deduce a numerical comparison between the force of gravity and the centrifugal force at the equator.

9. If a heavy body be whirled round in a vertical plane, and the centrifugal force at the top just keep the string extended; what will be the tension of the string at the lowest point of rotation?

10. In any orbit, let x = dist. p = perpendicular on the tangent:

law of the force in an ellipse round the centre, and in a circle with the centre of force in the circumference.

11. Deduce expressions for the chord of curvature passing through the focus, and the diameter of the curvature at any point of an ellipse.

12. All parallelograms described about any conjugate diameters of a given ellipse or hyperbola are of equal area.

13. Compare the centripetal and centrifugal forces at any point of an orbit; prove that in an ellipse round the centre, there are four points where these forces are equal.

14. Prove [Newton, Prop. XI.] that

Gv x vP: Qv2 :: CP2: CD2.

15. The perpendicular from the focus of a parabola upon the tangent is a mean proportional between the focal distances of the point of contact and the vertex.

16. Prove that the

force tending to the focus of a parabola ∞

1

D2

17. The velocity of a body revolving in a parabola round the the velocity of a body revolving in a circle at half the

focus distance.

18. If two bodies revolve in an ellipse in the same periodic time; one about the focus, and the other about the centre; compare the forces towards these centres at the extremities of the major axis, and find the distance from the centres at which the forces are equal.

19. If the force a

1

D2

and a body be projected in any direction,

except directly to or from the centre of force; prove that it will describe a conic section, and point out the relation between the velocity of projection and the particular curve described.

TRINITY COLLEGE, 1820.

1. (1). THE centripetal force (F) in any curve = Q.

being the perpendicular from the centre of force on the tangent, at distance (x).

Determine Q.

(2). Find the value of (F) in the ellipse-the force tending to the centre.

2. If a body be acted on by two forces tending to two fixed centres, it will describe, about the straight line joining those centres, equal solids in equal times.

3. A body describes a parabola about a centre of force situated in the focus:

(1). Find its position at any assigned time.

(2). Given two distances from the focus, and the difference of anomalies. Find the true anomaly.

4. The time of a body's descent, in a right line, towards a given

5. A body at P is urged by an uniformly-accelerating force in the direction PS, and at the same time is impelled in the opposite

1

direction by a force varying as

from S. Find its velocity at

(dist.)

any point N.

6. In the logarithmic spiral find an expression for the time of a body's descent from a given point to the centre, and prove that the times of successive revolutions are in geometrical progression.

7. A body acted on by a force varying as

1

(dist.)"

from the centre,

is projected from a given point, in a given direction, and with a

given velocity.

(1). Find the equation to the trajectory described.

(2). Determine in what cases the body will fall into the centre, or go off to infinity.