1

(dist.) 5

shew under what restrictions of

8. The force varying as the velocity of projection, the body's approach towards the 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.)s

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

from the

(dist.)2

(1). The point where the body will leave the spiral.

(2). The time of arriving at that point.

(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) [Prop. 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 assumed 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.

B x2'

(3). When it = Ax+ (x) being the distance.

TRINITY COLLEGE, 1823.

1. EXPLAIN the mode of reasoning by which Newton determines the ratios of quantities which vanish together; and prove that the ultimate ratio of the arc, chord, and tangent to each other is one of equality.

2. A body revolves in a semi-ellipse, and is urged by a force acting in lines perpendicular to the axis; find the law of force [Schol. to Prop. 8.]

3. Let P =

1

force on a body at distance r; u=- ; v = angle comprised between r, and a fixed line; t = time of motion; h=twice the area traversed by r in l", the following equations will obtain:

4. The angle between the distance and perpendicular increases or diminishes according as the velocity in the curve is greater or less than the velocity in a circle at the same distance.

1

5. Let a body which is acted on by a given force varying D be projected from a given point in a direction which makes with the initial distance D an angle = 8, and let velocity of projection = n times velocity from infinity,

the axis major of orbit described =

D

1

the axis minor..

n2

2nD. sin.d

√(1 — n2)

6. A body falls down the vertical axis of a paraboloid whose vertex is downwards; in what point of its descent will it have acquired a velocity sufficient to cause it to describe a circle along the surface of the paraboloid, if whirled round as a conical pendulum.

7. The perihelion distance of a Comet's parabolic orbit is a, and the radius of the Earth's orbit, which is circular, is r. The time during which the Comet is within the Earth's orbit is

where u intensity of force at distance unity.

8. If a body be projected from G towards a centre of force S,

1

which varies as and velocity of projection = n times velocity in

Do'

a circle at same distance, ne being less than 2; the time of describing GC

=

D

√{(2 — n2)3. v}

{2020'sin.20 sin.20'},

where DSG, ▲ ASD = 0, L ASG=0'. [Newton, Prop. 37.]

9. A body may be made to describe a lemniscata round a centre of force placed in the nodus, and whose intensity at distance

3h2a s

where a greatest value of r.

10. A body acted on by a repulsive force which varies

is projected from an apse with a velocity which is less than that acquired in falling from an infinite distance; construct the orbit. [Prop. 41, Cor. 3.]

11. A body is projected in such a manner as to describe a reci

procal spiral whose equation is

nth revolution =

12.

Determine the point of ultimate intersection of mn and pC [Prop. 44].

13. Let the force urging a body consist of two parts, one of which varies inversely as the square, and the other inversely as the cube of the distance; determine the orbit.

14. Explain why the theory of the ninth section cannot be applied to determine the motion of the apsides of the lunar orbit.

15. If the force vary as the distance, determine the nature and equation of the orbit, so that the times of descent from different points in it to the centre, may always be the same.

16. A body acted on by gravity is projected along the interior surface of a cylinder whose axis is vertical; define its motion.

17. Find the relative motion of several bodies which mutually attract each other with a force which varies as the distance.

18. The lunar months are longer in winter than in summer: explain fully the cause of this phenomenon.

19. The mean motion of the nodes is to the mean motion of the

p

apsides in a given ratio, and each varies as where = P's pe

riod round T, and P = T's round S.

p=

20. A particle of matter is situated in the axis of a paraboloid consisting of equal particles, each of which attracts with a force

proportional to the distance; find the magnitude and position of a sphere consisting of similar particles, which shall exert equal attraction on it.

21. A particle of matter rests any where on the surface of a solid of revolution, which revolves round its axis in a given time, and is attracted towards the centre with a force which is as some determinate function of the distance from it. Find the form of the solid and apply the result to the case when the force varies directly as the distance.

22. A body acted on by the uniform force of gravity moves in a resisting medium; the resistance to its motion in any point of which the

this result coincides with that of Newton, Book II. Prop. 10.

TRINITY COLLEGE, 1824.

1. STATE the method of reasoning by which Newton determines the ratio of quantities which vanish together. What objection has been raised against the principle, and how does Newton answer it? Shew that quantities which vanish together do not necessarily vanish in a ratio of equality.

2. Define similar curves, when referred to a point, and when referred to an axis: in each case shew that the corresponding sides, curvilinear, as well as rectilinear, are proportionals; and that the areas are in the duplicate ratio of the sides.

3. The ultimate ratio of the arc, chord, and tangent to each other is one of equality.

4. Every body that moves in any curve situated in a plane, and by a line drawn to a point, either at rest, or moving forward with uniform rectilinear motion, describes about it areas proportional to the time, is urged by a centripetal force tending to that point.

5. If h be twice the area traversed by the radius vector in l′′; prove the following expressions;