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

A

JM'

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 = 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

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.

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.

or,

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

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 "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 4ll' so as to touch both of them, its latus-rectum will be

1 ± l'

and the envelope of its directrix will be a conic, having the same focus and directrix as the given conics, and of which 4ll' the latus-rectum is 1 = 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."

The known property of a circle, that "two tangents make equal angles with their chord of contact," will be found, when transformed by the method now explained, to be equivalent to the theorem that "if two tangents be drawn to a conic from an external point, the portions of these tangents, intercepted between that point and their points of contact, subtend equal angles at the focus." From the fact that "all circles intersect in two imaginary points at infinity," we learn that "all conics, having a common focus, have a common pair of imaginary tangents passing through that focus." And, more generally, we may say that all similar and similarly situated conics reciprocate into a system of conics having two common tangents.

31. Two points, on a curve and its reciprocal, are said to correspond to one another when the tangent at either point is the polar of the other point. Two tangents are said to correspond when the point of contact of either is the pole of the other.

The angle between the radius vector of any point (drawn from the centre of reciprocation), and the tangent at that

point, is equal to the angle between the radius vector of, and tangent at, the corresponding point of the reciprocal

curve.

For, if P be the given point, PY the tangent at P, and S the centre of reciprocation, and SY be perpendicular to PY; and if P' be the pole of PY, and P'Y' the polar of P, then

P' will lie on SY, produced if necessary; and if SY' be perpendicular to P'Y', SY' will pass through P. Hence, since SP, PY, are respectively perpendicular to P'Y, SP', it follows that the angle SPY is equal to the angle SP'Y'.

32. We have investigated (Art. 10, Chap. IV.) the equation of the two tangents drawn to a conic from any given point (f, g, h). If in the right-hand member of that equation we substitute for 0, w (aa+bB+cy), w being an arbitrary constant, we shall obtain the general equation of all conics of which these lines are asymptotes. Now, since the asymptotes of the reciprocal conic with respect to (f, g, h), are respectively at right angles to the two tangents drawn from (f, g, h), it follows that the family of conics thus obtained will be similar in form to the reciprocal conic.

33. To find the co-ordinates of the foci of the conic represented by the general equation of the second degree.

Since the reciprocal of a conic with respect to a focus is a circle, it will follow from Art. 32 that the family of conics obtained as above must, if (f, g, h) be a focus, be circles also. Applying the conditions for a circle investigated in Art. 14, Chap. IV., it will be found that the terms involving & disappear of themselves, and our conditions assume the form

or

(Uh2 + Wƒ2 – 2V'hƒ) c2 + (Vƒ2 + Ug2 − 2 W'fg) b2

+ 2 (U' f2 + Ugh – W'hf — V'fg) bc = (Vƒ2 + Ug2 − 2 W'fg) a2 + (Wg2 + Vh2 – 2U'gh) c2

=

· (Wg2 + Vh2 − 2U'gh) b2 + (Uh2 + Wƒ2 − 2V'hƒ) a2

(Vb2+Wc2+2U'bc) ƒ2−2 (V'c+W'b) ƒ (bg+ch) +U (bg+ch)2 =( Wc2 + Ua2+2 V' ca) g2 −2 ( W'a+U'c) g (ch+af)+V(ch+af)2 =(Ua2+Vb2+2W'ab) h2 −2 (U'b+V'a) h (af+bg)+W (af+bg)2,