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 de scribed 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 = P perpen

x = dist

dicular on the tangent: centripetal force o

dp

́p3dx®

Apply this expression to determine the law of the

force in an ellipse round the centre, and in a circle with the centre of the 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 × vP : Qu2:: 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

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

1

19. If the force x and a body be projected

D2

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.

SPHERICAL TRIGONOMETRY.

1. EVERY plane section of a sphere is a circle. 2. The sum of the three angles of a spherical triangle is greater than two, and less than six, right angles.

3. The angles of a spherical triangle are A, B, C; the sides respectively opposite to them,

(a), (b), (c); the rad. of the sphere

the following theorems :

I. Cos. C =

= 1. Prove

cos. c-cos.a x cos. b

sin. ax sin. b

III. Tang.(A+B)=Cotan. C.

cos.(a-b)

cos.(a+b)

4. What are the general theorems deduced by the application of the formulæ I. and III. to the polar triangle ?

5. Prove Napier's rules for that case in which the complement of an angle is the middle part.

6. Given the obliquity of the ecliptic, the rightascension and declination of a star; find the angle of position in terms of those quantities.

7. In a spherical triangle, two sides and the in

cluded angle are given:-Required the third side in a form suited to logarithmic computations? 8. Find the area of a spherical triangle.

And the Freshmen had these

PROPOSITIONS IN PLANE

GEOMETRY.

1. If the exterior angle of a triangle be bisected by a straight line which also cuts the base produced, the segments between the bisecting line and the extremities of the base, have the same ratio which the other sides of the triangle have to one another. Shew that the converse is also true. 2. Equiangular parallelograms have to another the ratio which is compounded of the ratios of their sides.

one

3. The rectangle contained by the diagonals of any quadrilateral figure inscribed in a circle is equal to the sum of the rectangles contained by its opposite sides.

4. If the exterior angle of a triangle be bisected, and also one of the interior and opposite, the angle contained by the bisecting lines is equal to half the other interior and opposite angle of the

triangle.