For if on OA, OB, OC respectively (produced if necessary), we take points A', B', C', so that

OA.OA' = OB. OB' = OC. OC',

and through A', B, C' draw YZ, ZX, XY, severally at right angles to OA', OB', OC, then YZ, ZX, XY are respectively parallel to BC, CA, AB, or the triangle XYZ is similar and similarly situated to the triangle ABC.

We may observe further, that the point X, since it is the intersection of the polars of B and C, is itself the pole of the line BC, and therefore OX is perpendicular to BC, that is to YZ. Similarly, OY, OZ, are respectively perpendicular to ZX, XY. Hence, O is the intersection of the perpendiculars dropped from X, Y, Z on YZ, ZX, XY respectively. It may be convenient to call the point of intersection of the perpendiculars let fall from the angular points of a triangle on the opposite sides, the orthocentre of the triangle, or of its three angular points. Here we may say that "If a triangle be reciprocated with respect to its orthocentre, the reciprocal triangle will be similar and similarly situated to the given triangle, and will have the same orthocentre."

It will be seen by Art. 19, that any three points and their orthocentre, reciprocated with respect to any point S, give a quadrilateral, of which S is a focus.

26. If any conic be reciprocated with respect to an external point S, the angle between the asymptotes of the reciprocal hyperbola will be the supplement of that between the tangents drawn from S to the conic. (See Art. 9 of this chapter.)

Conversely, if an hyperbola be reciprocated with respect to any point S, we obtain a conic, which subtends at Ŝ an angle the supplement of that between the asymptotes of the hyperbola.

27. From the last article it follows that, if a parabola be reciprocated with respect to any point S on its directrix, we obtain a rectangular hyperbola, passing through S.

If a rectangular hyperbola be reciprocated with respect to

any point S on its circumference, we obtain a parabola whose directrix passes through S.

Again, if a conic be reciprocated with respect to any point on its director circle (i. e. the circle which is the locus of the intersection of two perpendicular tangents) we obtain a rectangular hyperbola.

If a rectangular hyperbola be reciprocated with respect to any point S not on the curve, we obtain a conic, whose director circle passes through S.

28. It is known that the conics passing through the four points of intersection of any two rectangular hyperbolas, is itself a rectangular hyperbola; and also that any one of these four points is the orthocentre of the other three. If, then, we reciprocate these theorems with respect to any one of the four points of intersection, we obtain the theorem that, "If a parabola touch the three common tangents of two given parabolas, its directrix passes through the intersection of the directrices of the two given parabolas, that is, through the orthocentre of the triangle formed by their common tangents." In other words, "If a system of parabolas be described, touching three given straight lines, their directrices all pass through the orthocentre of the triangle formed by the three given straight lines."

Again, reciprocating this system of rectangular hyperbolas with respect to any point S, we get, "All conics, which touch four given straight lines, subtend a right angle at either focus of the quadrilateral formed by these four straight lines." Or, in other words, "The director circles of all conics which touch four given straight lines, have a common radical axis, which is the directrix of the parabola which touches the four given straight lines.”

29. To find the polar reciprocal of a circle with respect to any point.

From what has already been shewn, we know that this will be a conic; we have now to investigate its form and position.

Let S be the centre of reciprocation, k the constant of reciprocation, MPM the circle to be reciprocated, O its centre,

MM' its diameter passing through S, p its radius, and let OS= c.

A

M S

M'

Fig. 19.

Through S draw any straight line cutting MPM' in P and Q.

On SPQ, produced if necessary, take two points I and Z, such that

SP.SY=SQ.SZ = k2.

The straight lines drawn through Y and Z perpendicular to SP will be tangents to the reciprocal conic.

which is constant. Hence, the reciprocal is a conic of such a nature that the rectangle under the distances from S of any two parellel tangents is constant. It is therefore a conic, of

which S is a focus, and of which the axis-minor is

2k2

(p2 - c2) It will be an ellipse, parabola, or hyperbola, according as p is greater than, equal to, or less than c, that is, according as

the centre of reciprocation lies within, upon, or without, the circle to be reciprocated. This agrees with what has been already shewn, Art. 9.

Let 2a, 2b, be the axes of the conic, 2l its latus-rectum, e its eccentricity.

To determine their magnitudes, we proceed as follows. The axis-major will be in the direction SO. Let A, A' be

its extremities.

1 SM + SM'__ 2p

Then

2 1
= +
ī SA SA

=

k2

=

Hence, l = or the latus-rectum is inversely propor

ρ

Thus the eccentricity varies directly as the distance of the centre of the circle from the centre of reciprocation, and inversely as the radius of the circle.

If d be the distance from S of the corresponding directrix,

or, the directrix is the polar of the centre of the circle.

30. We have now the means of obtaining, from any property of a circle, a focal property of a conic section.

Take, for example, Euc. III. 21. This may be expressed as follows: "If three points be taken on the circumference of a circle, two fixed and the third moveable, the straight lines joining the moveable point with the two fixed points, make a constant angle with one another." This will be reciprocated into (6 If three tangents be drawn to a conic section, two fixed and the third moveable, the portion of the moveable tangent intercepted between the two fixed ones, subtends a constant angle at the focus." This angle will be found, by reciprocating Euc. III. 20, to be the complement of one-half of the angle subtended at the focus by the portion of the corresponding directrix intercepted between the two fixed tangents.

Again, it is easy to see that "if a circle be described touching two concentric circles, its radius will be equal to half the sum, or half the difference, of the radii of the given circles, and the locus of its centre will be a circle, concentric with the other two, and of which the radius is half the difference, or half the sum, of the radii of the two given circles."

Hence we deduce the following theorem. "If two conics have a common focus and directrix, and their latera-recta be 21, 21', and another conic, having the same focus, be described so as to touch both of them, its latus-rectum will be

4ll'

は

l + l' ' and the envelope of its directrix will be a conic, having the same focus and directrix as the given conics, and of which the latus-rectum is

4ll' l F l'

Again, take the ordinary definition of an ellipse, that it is the locus of a point, the sum of the distances of which from two fixed points is constant. This is equivalent to "the sum of the distances from either focus, of the points of contact of two parallel tangents, is constant."

The reciprocal theorem will be, "If a system of chords be drawn to a circle, passing through a given point, and, at the extremities of any chord, a pair of tangents be drawn to the circle, the sum of the reciprocals of the distances of these tangents from the fixed point is constant."