23. It will be observed that the magnitude of the radius of the auxiliary circle affects the absolute, but not the relative, magnitudes or positions of the various lines in the reciprocal figure. As our theorems are, for the most part, independent of absolute magnitude, we may generally drop all consideration of the radius of the auxiliary circle, and consider its centre only. We may then speak of reciprocating "with respect to S" instead of "with respect to a circle of which S is the centre." S may be called the centre of reciprocation, k the constant of reciprocation.

24. As an example of the power of this method we will reciprocate the following theorem, "The three perpendiculars from the angular points of a triangle intersect in a point."

This may be expressed as follows: "If O, A, B, C be four points, such that OB is perpendicular to CA, and OC to AB, then will OA be perpendicular to BC."

Reciprocate this with respect to any point S, and the four points O, A, B, C give four straight lines, which we may call each by three letters abc, ab'c', a'be', a'b'c, respectively. Then, the fact that OB is perpendicular to CA is expressed by b and b' subtending a right angle at S, or by bsb' being a right angle. Again, the fact that OC is perpendicular to AB, shews that cSc is a right angle. Then the reciprocal theorem tells us that a Sa' is also a right angle. We may express this more neatly as follows: aa', bb', cc', are the diagonals of the complete quadrilateral formed by the four straight lines, hence it appears that at any point at which two of the diagonals of a complete quadrilateral subtend a right angle, the third diagonal also subtends a right angle. Or, in other words, The three circles, described on the diagonals of a complete quadrilateral as diameters, have a common radical axis.

The extremities of this axis may be conveniently called the foci of the quadrilateral.*

25. If the system formed by the four points O, A, B, C be reciprocated with respect to any one of them, O for instance, the triangle thus obtained will be similar, and similarly situated, to that formed by the other three points A, B, C.

* This name is proposed by Mr Clifford, in the Messenger of Mathematics.

RECIPROCATION WITH RESPECT TO A CIRCLE.

119

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 S 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,

RECIPROCATION OF A CIRCLE WITH RESPECT TO A POINT. 121

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

A

M S

M

O

Fig. 19.

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

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

SP. SY=SQ. SZ=k.

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 parallel tangents is constant. It is therefore a conic, of which S is a focus, and of which the axis-minor is

2k2

(p2 - c2) 3 It will be an ellipse, parabola, or hyperbola, according as is greater than, equal to, or less than c, that is, according as

P

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

Let 2a, 2b, be the axes of the conic, 27 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.

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.