12. Construct a dial on a plane inclined at any given angles to the meridian and horizon.

Of what nature is the error in the time shewn by a dial? What are the sources of the error? When does it vanish?

13. How does the projection of a small portion of the Earth's surface near the edge of the projection differ from that of the same portion near the centre, in the two projections commonly used?

ST. JOHN'S COLLEGE, MAY 1830.

1. In a spherical triangle where a, b, C are given, prove that c may be found from the formulæ

explain why these formulæ are inconvenient when the side c is nearly 180°, and investigate similar ones suited to that case.

2. If a, ẞ be arcs drawn from the right angle C of a spherical triangle respectively perpendicular to, and bisecting the hypothenuse; prove that

3. The hour-angle of a star, whose declination is 8, when seen in a certain vertical plane is h, and when next seen in the same vertical plane its hour angle is h': prove that tan. latitude of the place

4. The angle between the shadow on a horizontal dial, and the position it would occupy if the style were vertical, is at any time the complement of the angle between the vertical circle passing through

H

the Sun, and that which cuts his declination circle at right angles; shew that at the equinoxes this angle is the greatest at 3 o'clock.

5. Given the latitudes of two places, find the radius and position of the centre of the stereographic projection of the parallel passing through one, upon the horizon of the other.

6. The angular motion of the shadow of a vertical gnomon on a horizontal plane will be the greatest, when its length

a being the length of the gnomon, and the altitude of the Sun when on the prime vertical, on the given day.

7. How is it shewn that the diameter of the Earth's orbit subtends an evanescent angle at the fixed stars? If the fixed stars have an annual parallax, shew that the formulæ for aberration three months after the present time will represent its effects on their places.

8. The great circles in which a circumpolar star's motion seems to lie at three points of its course, are produced to meet, and the points of intersection are found to have the same latitude λ ; if ♪ be the declination of the star, prove that sin. obliquity

= ✓(sin. ( − a) — cos.2λ cos.28).

9. In a given latitude, and on a given day, find how long the Sun will be in rising, and where and when he will rise. Also determine that star which rises in a given point of the horizon at the same instant as the Sun's centre.

10. Describe the repeating circle, and its use in finding the angular distance between two objects by the method of crossed observations.

11. To calculate the places on the Earth's surface where a given solar eclipse is visible.

12. If a ship sail in the shortest course from a place in latitude 7, to a place in the equator, prove that in latitude l', where cot.l' = cot.2l+cosec.l, she will be nearer the end than the beginning of her course by a distance =

an

b

b ; a and b being the semi-axes of the

terrestrial spheroid, and n the length of the normal at starting, at which time also the ship's course is perpendicular to the meridian.

CORPUS CHRISTI COLLEGE, MAY 1830.

1. ENUMERATE the arguments by which the diurnal rotation of the Earth round its axis, and its annual motion round the Sun, are established.

2. Find the length of the normal at any point of the terrestrial meridian in terms of the latitude; and thence, by means of a seconds' pendulum determine the compression. The Earth being supposed to be an oblate spheroid of small eccentricity.

3. Define the Geodesic line, and shew that it is the shortest line that can be drawn on the Earth's surface between any two points of its course.

4. Explain the nature and use of the common vernier.

5. Given two altitudes of the Sun, and the time between; find the latitude and hour-angle.

6. Prove Brinkley's formula for the mean refraction, and shew that it is reducible to the same form as Bradley's.

7. The refraction being = m. tan.z + m'. (tan.z)3; give Lacaille's method of computing the coefficients.

8. Find the effects of parallax on the hour-angle, and declination of a heavenly body.

9. Define the terms, Precession and Nutation, and find the effects of the former on the right ascension and declination of a given

star.

10. Find the longitude of the perihelion, and the time of the Earth's passing through it.

12. Supposing a Comet to move in an orbit of great eccentricity;

570520

investigate the following formula for determining the distance (g) of the Comet from the Sun, corresponding to given anomalies;

sin.0.sin..sin.20.sin.2p+ .sin.30.sin.30 - &c.

where 0. sum of the true and eccentric anomalies, and = their difference.

13. Define the equation of the centre, and shew that in the Earth's orbit, its greatest value is nearly equal to twice the eccentricity.

14. Given three distances of a planet from the Sun, and the corresponding arguments of latitude : find the place of the perihelion, and the true anomaly at the first observation.

15. Explain the phases of Venus. State also M. Schroeter's reasons for supposing that her atmosphere is much denser than that of the Moon.

16. Find the time, magnitude, and duration of a lunar eclipse. 17. The orthographic projection of the rhumb line on the plane of the equator is an hyperbolic spiral.

18. Construct an horizontal dial for a given latitude, and determine the position of the last hour-line.

19. Given the Sun's altitude; find at what angle a straight rod must be inclined to the horizon, that the length of its shadow may be the greatest possible.

20. Describe fully the lunar method of determining the longitude at sea. Give also the analytical process adopted by Borda, for determining the true distance.

21. Given, tan.

<-1

101-4711276,

tan: 111.4801364,

tan. 1 13 1.4940244,

find tan. 11.63 by the method of interpolations:

22. Find the time of year when the duration of twilight is

shortest.

23. Whence does it appear manifest that the Moon is surrounded by a much rarer atmosphere than that of our Earth? From what circumstance did some of our astronomers formerly deny the existence of an atmosphere at the Moon?

24. Supposing the Earth's orbit to be a logarithmic spiral, find the equation to the aberratic curve.

25. Explain the method of least squares as given by Gauss; and show that it gives the most probable values of the unknown quantities.

26. Determine the Sun's parallax by means of a transit of Venus over the Sun's disk.

1. Fx

D3

CAIUS COLLEGE, MAY 1830.

Find the time of a body descending in a straight

line from a given point to the centre of force.

A body is projected from any point in any direc

D2

tion with a velocity equal to that from infinity. Find the equation to the curve described, the position of the apse, and the whole angle described before the body falls into the centre.

3. A body is revolving in an ellipse, round the focus, and at the extremity of the axis-minor half the force is suddenly taken away; determine the alteration which takes place in the form of the orbit.

4. Prove that the force by which the cissoid may be described

round a centre of force in the cusp varies as

(cosec.0)2
r$

angle which the radius vector (r) makes with the axis.

where is the

5. Find the time of an oscillation in a circular arc, when the

force in the direction of the curve varies as

from the lowest point.

Ө

sin.0'

being the distance

6. A body is describing a circular arc, and acted upon by a force tending to a given fixed point without the circle; determine its velocity at any point.

1

D2

7. The particles of a cone attracting with a force which varies as

, compare the attraction on a particle in the vertex with that on

a particle in the centre of the base.