1849.

(A). Find the law of force under the action of which a body may describe an ellipse, one of the foci being the centre of force.

(B). If v, v' be the velocities at the extremities of any focal chord, and u that at the extremity of the latusrectum, then will v2, u2, v", be in arithmetical progression.

The velocity at any point of the ellipse is that due to of the chord of curvature through the focus;

Similarly, if v' be the velocity at the other extremity of the focal chord PSP',

In the same manner, if u is the velocity at the extremity of the latus-rectum and L the semi-length of the latus-rectum,

But by a well-known property of the ellipse,

or v2, u2, v22, are in arithmetical progression.

EXAMPLES FOR PRACTICE.

(A). Book 1. prop. 26.

(B).

(A).

(B).

(A).

(B).

(A).

(B).

(A).

(B).

EUCLID.

The sides AB, AC of a right-angled triangle, in which A is the right angle, are produced; if the lines bisecting the exterior angles meet in O, and perpendiculars be drawn to the sides produced, shew that the figure OMAN is a square.

Book I. prop. 32.

A semicircle ABDC is trisected by the straight lines OB, OD drawn from the centre 0; shew that the line joining B, C bisects OD.

Book 1. prop. 34.

Shew that any straight line passing through the middle point of the diameter of a parallelogram bisects the parallelogram.

Book 1. prop. 39.

Two straight lines AC, BD, cut in E. If the triangle ABE equal the triangle CED, and the triangle AED equal the triangle BEC, the figure ABCD is a parallelogram.

Book III. prop. 11.

If two circles touch each other internally, and any circle be described touching both, prove that the sum of the distances of its centre from the centres of the two given circles will be invariable.

(A).

(B).

(C).

Book III. prop. 22.

If all the angles of a quadrilateral inscribed in a circle are bisected by the diagonals, it must be a square.

If circles be described about the four triangles formed by the intersection of four straight lines, shew that these circles all pass through one point.

(A). Book III. prop. 26.

(B). If a circle be described about a triangle ABC, and perpendiculars be let fall from the points A, B, C on the opposite sides of the triangle, and be produced to meet the circle in D, E, F, respectively; shew that the arcs EF, FD, DE are bisected in A, B, C.

(A). Book III. prop. 31.

(B).

(A).

(B).

(A).

(B).

(C).

Two equal circles cut one another in A and B; if the diameters AC, AD be drawn in the two circles, shew that CB, BD are in the same straight line. Also, if the diameter DA of one centre be produced to meet the other in E, shew that E is a point in the circle described with centre B and radius BD.

Book III. prop. 32.

A point A is taken in a circle such that, if the tangents AB, AC be drawn to an equal circle, and be produced backwards to meet the former circle in D and E, the chord DE BC. Shew that the triangle ABC is equilateral. =

Book III. prop. 36.

ABCD is a quadrilateral inscribed in a circle, such that the sides AB, DC produced, and the sides AD, BC produced, meet respectively in two points E, F of a concentric circle. Shew that EF cannot be parallel to BD unless each of the angles ABC, ADC are right angles.

If two circles cut one another, then the common chord produced bisects their common tangent.

(A). Book IV. prop. 15.

(B).

(A).

(B).

(C).

(A).

Six equal circles pass through one point, so as by their mutual intersections to determine the angular points of two regular. hexagons. Shew that of the two circles circumscribing these hexagons, one is equal to any one of the given circles, and the area of the other is three times the àrea of any one of them.

GEOMETRICAL CONIC SECTIONS.

The tangent at any point of a parabola makes equal angles with the axis and with the line joining the point with the focus.

If the diameter at the point P in a parabola be produced to meet the directrix in M, and MS be drawn to the focus S, then the perpendicular from P on MS will be a tangent at P.

Draw a pair of tangents to a parabola from a given point in the directrix.

If from the focus (S) of a parabola whose vertex is A, SY be drawn perpendicular to the tangent PT, prove that AY is the tangent at the vertex.

(B). In SP a point A' is taken so that SA' SA; shew that SA'.A'PAY".

(A).

(B).

(A).

(B).

Define an ellipse.

Supposing no bodies to exist in space but the sun and a small plane mirror which moves so as always to be a tangent to its path, find the locus of the sun's image.

The rectangle under the abscissæ of the axis-major of an ellipse is to the square of the semi-ordinate as the square of the axis-major to the square of the axis-minor (AN.NM: PN2 :: AC2 : BC”). :

Produce NP to meet the auxiliary circle in Q; draw PR parallel to QC, meeting the axis-major in R. Shew that PR

BC.