the circumference of the other. If BC, BD be joined, shew that AB, or AB produced if necessary, bisects the angle CBD.

3. Draw a line to touch one given circle, so that the part of it contained by another given circle shall be equal to a given straight line not greater than the diameter of this latter circle.

4. Reduce to its simplest form the expression

5.

(1 − a2) (1 − b2) (1 − c2) − (c + ab) (b + ca) (a + bc)

Find a whole number which is greater than three times the integral part of its square root by unity. Shew that there are two solutions of the problem and no more.

6. If p - a, p, 4 + a be three angles whose cosines are in harmonical progression, shew that

7. A person, wishing to ascertain his distance from an inaccessible object, finds three points in the horizontal plane at which the angular elevation of the summit of the object is the same. Shew how the distance may be found.

8. Draw a parabola to touch a given circle in a given point, so that its axis may touch the same circle in another given point.

9. If a circle be described touching the axis major of an ellipse in one of the foci, and passing through one extremity of the axis minor, the semi-axis major will be a mean proportional between the diameter of the circle and the semi-axis minor.

10. If AB, CD, two lines in an ellipse, not parallel to each other, make equal angles with either axis; the lines AC, BD and AD, BC will also make equal angles with either axis.

11. Two forces F and F", acting in the diagonals of a parallelogram, keep it at rest in such a position that one of its edges is horizontal; shew that F sec a = F' sec a' = WV cosec (a + a'), where W is the weight of the parallelogram, a and a' the angles between its diagonals and the horizontal side.

12. A quadrilateral figure possesses the following property; any point being taken and four triangles formed by joining this point with the angular points of the figure, the centres of gravity of these triangles lie in the circumference of a circle: prove that the diagonals of the quadrilateral are at right angles to each other.

13. If the angle of a hollow cone polished internally be any submultiple of 180o, a cylindrical pencil of rays incident parallel to the axis will after a certain number of reflexions be a cylindrical pencil parallel to the axis, and of the same diameter as the incident pencil.

14. A cubical box is half filled with water and placed upon a rough rectangular board, so as to have the edges of its base parallel to those of the rectangle; if the board be slowly inclined to the horizon, determine whether the box will slide down or topple over.

15. A body floats in a mixture of two given fluids with a volume A immersed; one-half of the mixture being removed, and its place supplied by an equal quantity of the lighter fluid, the same body floats with a volume A + B immersed. Determine the ratio of the quantities of fluid in the original mixture, supposing the volume of the mixture to be equal to the sum of the volumes of the component fluids.

Explain the result when the densities of the fluids are as A+B to A - B.

16. There are two walls of equal known height at right angles to each other, and running in known directions; shew how to find the Sun's altitude and azimuth by observing the breadth of the shadows of the two walls

at any given time. And prove that the sum of the squares of the breadths of the shadows will be the same whatever be the direction of the walls.

17. If the same two stars rise together at two places, the places will have the same latitude. And if they rise together at one place and set together at the other, the places will have equal latitudes, but one North and the other South.

18. A body is projected from a given point in a horizontal direction with given velocity, and moves upon an inclined plane passing through the point. If the inclination of the plane vary, find the locus of the directrix of the parabola which the body describes.

19. An imperfectly elastic ball lies on a billiard-table, determine the direction in which an equal ball must strike it in order that they may impinge upon a side of the table at equal given angles.

20. The circle described through two points of an equiangular spiral and the point of intersection of the tangents at those points will pass through the pole. Prove this, and apply the proposition to shew that the curvature at any point of an equiangular spiral varies inversely as the distance of the point from the pole.

21. A bead running upon a fine thread the extremities of which are fixed describes an ellipse in a plane passing through the extremities, under the action of no external force; prove that the tension of the thread for any given position of the bead is inversely proportional to the square of the conjugate diameter.

22. The centres of two equal spheres (elasticity e, radius r,) move in opposite directions in a circle (radius R) about a centre of force varying inversely as the square of the distance; determine the motion of the spheres after they have impinged, supposing that e = ; and prove

R2

that the latus rectum of the conic section described after

the second impact will be 2e R.

1850.

MODERATORS.

LEWIS HENSLEY, M.A., Trinity College.
JOHN SYKES, M.A., Pembroke College.

EXAMINERS.

WILLIAM BONNER HOPKINS, M.A., St Catharine's Hall.
ARCHIBALD SANDEMAN, M.A., Queens' College.

THURSDAY, Jan. 3. 9...12.

1. THE opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is, divides them into two equal parts.

If the opposite sides, or the opposite angles of any quadrilateral figure be equal, or if its diagonals bisect one another, the quadrilateral is a parallelogram.

2. Describe a square which shall be equal to a given rectangle.

Given a square and one side of a rectangle which is equal to the square, find the other side.

3. In a circle, the angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

The greatest rectangle that can be inscribed in a circle is a square.

4. Cut off a segment from a given circle which shall contain an angle equal to a given rectilineal angle.

Divide a circle into two segments such that the angle in one of them shall be five times the angle in the other. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

5.

Shew that the base of the triangle is equal to the side of a regular pentagon inscribed in the smaller circle of the figure.

6. Find a third proportional to two given straight lines.

AB is a diameter, and P any point in the circumference of a circle; AP and BP are joined and produced if necessary: if from any point C of AB a perpendicular be drawn to AB meeting AP and BP in points D and E respectively, and the circumference of the circle in a point F, shew that CD is a third proportional to CE and CF.

7. If two straight lines be at right angles to the same plane, they shall be parallel to one another.

8. If a line be drawn bisecting the angle between the focal distance of any point of a parabola and a perpendicular from that point upon the directrix, every point of the line will lie outside the parabola.

9. The perpendiculars from the foci on the tangent of an ellipse intersect the tangent in the circumference of a circle having the axis major as diameter.

Employ this proposition to find the locus of the intersection of a pair of tangents at right angles to each other. 10. The rectangle under the abscisse of any diameter of an ellipse is to the square of the semiordinate, as the square of the diameter is to the square, of the conjugate. (PV. VG: QV2 :: CP2: CD2).

11. In the hyperbola the rectangle under the lines intercepted between the centre and the intersections of the axis with the ordinate and tangent respectively is equal to the square of the semiaxis major.

Through N draw NQ parallel to AP to meet CP in Q; prove that AQ is parallel to the tangent at P.

12. If a point K be taken in the major axis of the hyperbola such that CK is a third proportional to CS and CA, and a perpendicular to the axis be drawn through K, the distance of any point in the curve from this line will bear a constant ratio to its distance from the point S.