centre, and its recess towards infinity, will be limited by asymtotic circles.

9. The difference of the forces by which a body may be made to move in the quiescent and in the moveable orbit varies as from the centre.

1

(dist.)3

10. (1) Deduce the equation to the orbit in fixed space.

(2) Shew that when any one of Cotes's three last spirals is made the moveable orbit, the orbit in fixed space will be one of the same species.

11. Why are the principles of the 9th section inapplicable to the complete explanation of the planetary motions?

12. Make a body oscillate in a given hypocycloid.

13. Given the position of a body on a rigid logarithmic spiral

which it is made to describe by a force varying as

pole, find

1

(dist.)2

(1) The point where the body will leave the spiral.
(2) The time of arriving at that point.

from the

(3) The elements of the orbit which it will then describe.

14. Demonstrate the 66th Proposition.

15. Find expressions for the disturbing forces on P when at a given distance from quadratures.

16. If ST and the absolute force of S be changed, the periodic

linear errors of P vary as

1

(Period of T)

Pr. 66. Cor. 14.

17. Prove that the mean disturbing force on the moon in a whole revolution = - the mean addititious force.

18. When the force varies as the (dist.), shew that there will be no disturbance.

(1) Had this law pervaded the universe, what would have been the consequence?

(2) On what circumstances in the variations of the elements of the orbits does the stability of the planetary system depend?

19. S is the centre, SA the radius of a sphere, each of whose

particles has an attractive force varying as

1

(dist.)**

Having as

sumed in SA produced any point P, and having taken SP: SA :: SA SI, find the ratio of the attractions which the whole sphere exerts on equal corpuscles placed at P and I.

20. The attractions of ellipsoids upon particles placed on the surface urging them in directions perpendicular to any principal section are proportional to the distances of the particles from that section.

21. Prove that a shell of homogeneous matter contained between

two concentric spherical surfaces, will attract a particle placed without it in the same manner as if all the matter were collected in the centre, in the following cases:

(1) When the law of attraction is that which obtains in nature. (2) When it varies as the distance.

1. EXPLAIN what is meant by continued finite curvature. Shew that if QP be any arc of a curve, and QR a subtense perpendicular to the tangent, limit

QP2

QR

diameter of curvature at the

point P; and apply this expression for finding the diameter of curvature at the vertex of a cycloid.

2. Let AB be any arc of a curve of finite curvature, AK, BK normals at A and B meeting in K, and BG perpendicular to the chord AB meeting AK in G; prove that in the limit AK AG :: 1: 2.

3. Investigate the relation between the centripetal and centrifugal forces at any point in any orbit: the equation to the curve in which they are equal; and the law of the force by which it will be described.

1

4. If the force c and a body descend in a straight line;

D2

find the velocity and time corresponding to any given space by Newton, Prop. 39. Cor. 2, 3.

5. If a body be projected in any direction from a given point above a given plane, and be acted upon by a force perpendicular to the plane, and varying inversely as the nth power of the distance from it; find the equation to the trajectory, and shew for what values of n, that equation will be expressed in finite terms.

6. A body begins to fall from A to a centre of force S, varying inversely as the cube of the distance; find the nature of the curve

AP, when the time down AN is equal to the time of describing NP with the velocity acquired at N.

7. Find generally the equation to the orbit in fixed space in Sec. 9, and from that equation, shew that the difference of force in the fixed and moveable orbit varies as

1

D3

8. If the force vary as A", shew that the angle between the apsides in orbits nearly circular =

√n+3

nearly, and when n=1

explain the reason why we obtain an accurate result.

9. Find the nature of the curve which by its rotation round its axis will generate a surface, in which the times of revolution in circles parallel to the horizon shall be equal at all altitudes.

10. If a string will just bear (p) pounds; through what angle must it be made to oscillate with a weight (q) less than (p) at its extremity so that it may all but break?

11. Investigate an expression for the tangential force in P's orbit supposed circular, and find the velocity generated by it from quadrature to syzygy.

12. Find those positions of the apse of P's orbit where the excentricity is a maximum and minimum, and explain fully Newton's reasoning in Cor. 9. Pr. 66.

QUEEN'S COLLEGE.

MAY, 1819.

MISCELLANEOUS PROBLEMS.

1. GIVE a definition of force; and distinguish between accelerating and moving force.

2. The sum of two forces is to the compound force, when they act at an angle (0) as (m) to (n), shew that the angle (8), which the compound force makes with either of the other two, may be determined from the equation

3. Two weights P and Q are supported upon two inclined planes AB, BC, of given elevation, by a string passing over a pulley at D.

Find the position of equilibrium, and shew that if the system be put in motion P Q :: vel of Q: vel' of P; the velocities being measured in a direction perpendicular to the horizon.

4. AD is horizontal, DC vertical, Q a weight connected with one extremity of a beam AB, by a string passing over a pulley at C, in such a manner that CB is vertical. Find Q when there is an equilibrium, and shew that the equilibrium will be maintained, whatever be the position of the beam AB; CB remaining vertical.

5. AB is a vertical line. Find the nature of the curve traced out by C, such, that the square of the time down AC + the square of the vely down AC is a constant quantity.

6. P and Q are two weights connected by a string passing over a fixed pulley, whereof P is the greater: at the end of (t") an additional weight (g) is annexed to Q. Find the velocity of P, after any assigned time.

7. If a ball whose elasticity is to perfect elasticity as (n) to (1), impinge upon a perfectly hard plane; shew that tan. I tan. R::n : 1, I and R being the angles of incidence and reflexion.

8. If (0) be the angular distance of a body from the lowest point in a circular arc; shew that the force in the direction of the arc is to that in the direction of the chord as 2 cos.

0

2

04

: 1.

9. If a ball whose elasticity is to perfect elasticity as (n) to (1) be projected in vacuo at an angle (0) with vel" (V); prove that the sum of all the ranges will be expressed by

V2 sin. 26

g. (1−n)

g being

taken to represent gravity, or 321 feet.

1. Divide a given cylinder, containing two fluids of different specific gravities, which will not mix, into two parts, such that the pressures on each shall be equal, the depth of each fluid being given.

2. If a plane be immersed vertically in a fluid of which the density varies as the nth power of the depth; find the pressure upon the whole plane.

3. Having given the specific gravities of wood and water, find the specific gravity of brass when (a) cubic inches of brass connected with (6) cubic inches of wood will just float.

4. Explain the reason why a cannon ball will fly farther if a motion of rotation be impressed upon it in a direction perpendicular to the line of its motion, than if that rotatory motion took place in the direction of its motion.

5. How long will a cylinder be in emptying itself by means of wo orifices of given dimensions; one in the bottom, and another at a given distance from the bottom,

6. Find the diameter of a capillary tube.

7. If altitudes be taken above the earth's surface in arithmetical progression, prove that the altitude of the barometrical column will decrease in geometrical progression.

8. There are two barometers, in one of which was left (m) inches, and in the other (n) inches of air. In consequence of a change in the state of the atmosphere, the former fell (d) inches. Find the corresponding variation in the latter.

9. If a body descend in a medium whereof the resistance varies as the (vel'); find the space described, corresponding to any. assigued time.

1. A ray of light incident upon the concave surface of a spherical reflector, in a direction parallel to the axis, after two reflections intersects the incident ray in a given point; find the angle of incidence.

2. In what point of the horizontal line AB will a given line (C), in the same plane with AB, appear the greatest?

3. If (R) and (r) be the radii of the surfaces of a double convex lens, what alteration must be made in (r), so that the focal length may be increased (p) times?

4. Find the focal length of a double concave lens from these data-the distance of the image from the lens, and the ratio of the object to the image.

5. A cylindrical vessel of given dimensions is so situated, that an eye can just see the farther extremity of the diameter. How much water must be poured in, so that the eye may just see a (cth) part of it?

6. If a quadrant be immersed vertically in a fluid, one of its radii being coincident with the surface; find the equation to the image.

7. Determine the vertical angle of an isoceles glass prism, such that rays incident perpendicular to one side may just be reflected by the base perpendicular to the other.

8. If the Sun's light be transmitted through a prism, a coloured image of the Sun will be formed on the opposite wall; and if the prism be turned round its axis, this image will ascend and descend alternately. Shew that the refractions on each side of the refracting angle are equal at the instant the image ceases to ascend, and begins to descend.