DEMONSTRATIONS OF SOME PROPERTIES

OF THE CONIC SECTIONS.*

1. THE position of the circle of curvature at any point of a Conic Section, may be readily determined by the following construction. Describe a circle touching the curve at the given point and cutting it in two others, then the chord in the conic section which passes through the given point and is parallel to the line joining the two points of section, is a chord of the circle of curvature at the given point.

Taking the origin at the point in the curve, the equation to the conic section is

y2 + Bxy + Сx2 + Dy + Ex=0.

If we make the normal the axis of x and the tangent that of y, when x=0 the two values of y must also = 0, and therefore D=0, which reduces the equation to

y2 + Bxy + Cx2 + Ex=0.

The equation to a circle touching the curve at the given point, that is, being also a tangent to the axis of y at that point, is

y2+x2+ E'x=0.

At the intersection of the curves we may combine the equations in any manner; subtracting them we have

Bxy + ( C − 1) x2 + (E− E') x = 0,

which gives x=0, By+ (C−1) x + E — E' = 0.

* Cambridge Mathematical Journal, Vol. I., p. 61.

The first of these equations gives the origin, the second gives a relation between the coordinates of each of the points of section: and as it is linear it is the equation to the straight line joining them. Now the angle which this line makes with the axis of x is determined by the ratio 1-C

B; and as this is independent of the circle, the line will

В

remain parallel to one position, as the circle varies in size. But when the circle becomes the circle of curvature, one of the points of intersection coincides with the point of contact, and the line joining the points of intersection, which remains parallel to one position, must pass through the point of contact. Therefore, if from this point a line be drawn parallel to a known position of the line of intersection, it will pass through the point in which the circle of curvature cuts the curve, and will thus be a chord both in the curve itself and in the circle of curvature. Knowing now one chord of the circle of curvature, and the position of its diameter, which coincides with that of the normal, we can determine the circle altogether.

2. The following is another and a very curious method of determining the centre of curvature in the Conic Sections. It was first given by Keill, but seems to have been rather strangely neglected by the subsequent writers on this subject.

From any point P (fig. 2) in the curve draw the normal PN cutting the axis in N; at the point N draw NQ perpendicular to the normal, and meeting the focal chord through P in Q. From Q draw 20 perpendicular to the focal chord and meeting the normal in O; then O is the centre of the circle of curvature at the point P. Draw FY from the focus perpendicular to the tangent, and let

FP=r, FY=p, PN=N.

In the right-angled triangle PQO, we have plainly

Also in a conic section we have, if R be the radius of cur

and O is the centre of the circle of curvature at P.

3. In the Cambridge Transactions, Vol. III., Mr. Morton has demonstrated a number of curious properties of the Conic Sections in relation to the generating cone; but he does not seem to have noticed the following one. If a sphere be described round the vertex of a cone as centre, the latera recta of all sections of the cone made by planes touching the sphere are equal. Taking the vertex of the cone as the origin, and the axis of the cone as the axis of x2 the equation to the cone will be

x2 + y2=m*z*.

And if we suppose the cutting plane to be perpendicular to the plane of xz, its equation will be

z cosa+x sina=r;

where r is the radius of the sphere, and a the angle which the perpendicular from the origin on the plane makes with

the axis of z.

we get

Eliminating between the two equations,

x2 (cos2 a — m2 sin2 a) + y2 cos2 a +2m2rx sin a= m2r2, which is the equation to the projection of the section on the plane of xy. This equation, which is that to an ellipse, is not referred to its centre, but if we so refer it, it becomes

sin2a)2

x2 (cos2a - m2 sin2 a)2 + y2 cos2 a (cos2a — m2 sin" a) = m2r2. cos2 a. Now, if a', b' be the axes of the projection,

If a, b be the axes of the section, as the cutting plane is perpendicular to the plane of xz, and makes an angle a with the plane of xy,

And for the latus rectum,

a' = a cosa,

b' = b.

which being independent of a, is the same for all sections for which r is the same; that is, for all those which are made by planes touching the sphere.

From this it appears, that the latus rectum is equal to the diameter of the sphere multiplied by the tangent of half the vertical angle of the cone.

SOLUTIONS OF SOME PROBLEMS IN

TRANSVERSALS.*

THE name of Transversals was given by Carnot to lines considered in their relations of mutual intersection. Many of their properties are very curious, and form interesting problems in Analytical Geometry, though it was not in this way that Carnot considered them. His method was, to proceed step by step from the more simple properties to the more complicated: but it seems better to consider each independently.

1. The following problem, under a slightly different form, was given in one of the Problem Papers for 1836.

If two lines AB, CD intersect in O so that AB is bisected, and if the lines AC, BD meet, when produced, in E, and AD, BC in F, then the line EF is parallel to AB.

Take O as the origin, and AB, CD as the axes of x

and y.

Let OB=a, OA=-a, OD=b, OC=-c.

The equations to BD and AC are

At their intersection the equations may be combined in any manner; therefore, subtracting them,