pendicular to its conjugate of the pencil p'. For, representing the pole of the axis aa, which lies at an infinite distance, by the letter A, we have the three rays aa, pA, and Sp of the first pencil conjugate respectively to the perpendiculars p'A, aa, and Sp' of the second pencil; so that the pencils p and p' generate a circle whose diameter is pp', and any two conjugate rays of these two pencils are perpendicular to each other. Hence

COR. I.-The points p and p', in which any two conjugate rays of this class cut the axis aa, are in involution; and, therefore, the system of pairs of points in which such rectangular conjugate rays meet the axis aa is an involute system.

COR. II. If the punctuated line in involution has two double points they will coincide with the foci of the curve. In fact any two rectangular rays issuing from a focus are conjugate.

COR. III. If the punctuated line has no double point, each of the two points, whence it is projected by a rectangular pencil, is a focus of the curve. In the latter case the foci are situate on the axis of the curve perpendicular to aa, and form the double points of an involution on that axis, formed in the same way as the involution on the axis aa.

COR. IV. The foci of an ellipse or a hyperbola are symmetrically situate with respect to the centre of the curve. In fact the foci are the double points of the involute line aa, and the centre is the conjugate of the point at infinity. Hence (§§ 19 and 27) the foci lie at equal distances on opposite sides of the centre, and the foci are harmonically separated by any two rectangular conjugates. In the case of the parabola the second focus lies at infinity, since the tangent at infinity touches the axis in a point of which it is the polar. Wherefore this is the second double point or focus of the curve. The distances from this

second focus to any two conjugate points are equal. Similarly, the first double point bisects the distance between any two conjugate points, inasmuch as double points or foci are harmonically separated by pairs of conjugate points; and, if then the fourth point passes to infinity, the second point will bisect the distance between the first and third.

29. Let ƒ and ƒ' (Fig. 50) be two foci of a curve of the second order, of which one lies at an infinite distance upon the axis when the curve is a parabola; then any pair of rectangular rays Sp and Sp' are harmonically separated by the rays Sf and Sf'. But Sp is perpendicular to Sp'; therefore, since in any harmonic pencil two rectangular conjugate rays bisect the adjacent angles contained between any other two conjugate rays, Sp and Sp' bisect the angles formed by Sf and Sf'. If S is a point on the curve, one of the two lines Sp and Sp' is tangent to the curve.

If S lie outside of the curve, Sp and Sp' bisect also the angles between the tangents drawn from S to the curve; for the intersection of the line of contact with the polar Sp is harmonically separated from the pole by the two points on the curve. Thus every tangent to a curve of the second order forms equal angles with the two lines joining its point of contact to the foci; and the angle formed by two tangents drawn from an external point to a curve of the second order is bisected by the same line as the angle formed by the rays drawn from the point to the foci. In the case of the parabola the line joining the second focus. and the external point is parallel to the axis.

DEF. The directrix of a conic is the polar of its focus. 30. The segment of a tangent comprised between the point of contact and the directrix is projected from the focus under a right angle. For the two rays of projection are conjugate, since the point of intersection of the tangent and directrix has for its polar the line joining the focus to the point of

contact, whilst the point of contact is the pole of the

31. The line joining the real focus of a conic to the point of intersection of a pair of tangents bisects the angle formed by the rays joining the same centre to the points of contact. Let TA and TB be any pair of tangents to a curve of the second order, so that AB is the polar of the point T (Fig. 51). Since F is the pole of the directrix bp

and T is the pole of AB, we have p, the intersection of the polars bp and AB, as the pole of TF; hence the rays FT and Fp are conjugate and therefore at right angles. Moreover, since p is the pole of FT, the range pAT'B is harmonic, so that the pencil FpAT'B is harmonic. But, as we have already shewn, FT' and Fp are rectangular conjugate rays and so bisect the adjacent angles formed by FA and FB.

32. If in Fig. 51 the lines Bb and Aa be drawn parallel to FT meeting the directrix in b and a respectively, the points a and b will be harmonically separated by t and p; since the range pAT'B is projected from infinity upon the line bp in path.

Again, the angles BFT' and FaA are each equal to FBb. But the angle

AFT'=BFT'=F6B,

Thus, the triangles aAF and bBF being similar,

or, if Bb' and Aa' be drawn perpendicular to the directrix,

FA FB
AaBb'

so that the distances of any point on a conic from the focus and the directrix bear to each other a constant ratio. In the case of the parabola the value of this ratio is unity; since the vertex is harmonically separated from the point at infinity by the focus and the directrix. In the case of the ellipse or the hyperbola the ratio is the same for both foci, each being referred to the corresponding directrix. If then r and be the focal distances of any point on the curve; d and d' the distances of the same point to the two directrices, we have

wherefore

r

;=a constant ;

But the distance d+d' in the case of the ellipse, and d - d' in the case of the hyperbola, represents the constant distance between the two directrices. Hence rr' is also constant and equal to the length of the major axis.

33. The feet of all perpendic ulars let fall from the foci of a central conic upon the tangents

M

Fig. 52.

are situate upon the circumference of a circle whose diameter is equal to the major axis. Let F and F' (Fig. 52) be the two foci of an ellipse; A and A' the ends of any diameter. The quadrangle FAF'A' is a parallelogram; since its diagonals bisect each other in the centre of the curve at M.

Moreover, the sides FA and F'A form equal angles with the tangent at A (§ 29); wherefore, since A'F' is parallel to AF and intersects the tangent at A in K, the triangle AFK is isosceles. The foot N of the perpendicular let fall from the focus F' upon the tangent bisects AK; the point M is the centre of AA'; wherefore MN is parallel to A'K and equal to A'K. Again,

whence

and

N

S

N

34. The feet of all the perpendiculars let fall from the focus

B

Fig. 53.

F'

of a parabola upon the tangents are situate upon the tangent at the vertex (Fig. 53). For by § 29 the angle formed by the tangents drawn from any external point N to the curve has the same line of bisection Nx as the angle formed by the rays NF' and NF drawn from N to the foci. Hence, if N be the point of intersection of the tangent at any point A and that at the vertex S, we have

xNA=xNS,

But F'NS is a right angle, therefore FNA is likewise a right angle.

35. The preceding two propositions furnish the means. for finding foci of a given conic. But the following method