form S-kTT', where T and T' represent the tangents at these points.

253. If the line a be parallel to an asymptote of the conic S, it will also be parallel to an asymptote of any conic represented by S-kaß, which then denotes a system passing through three finite and one infinitely distant point. In like manner,

if in addition ẞ were parallel to the other asymptote, the system would pass through two finite and two infinitely distant points. Other forms which denote conics having points of intersection at infinity will be recognized by bearing in mind the principle (Art. 67) that the equation of an infinitely distant line is 0.x +0.y +C=0; and hence (Art. 69) that an equation, apparently not homogeneous, may be made homogeneous in form, if in any of the terms which seem to be below the proper degree of the equation we replace one or more of the constant multipliers by 0.x +0.y+C. Thus, the equation of a conic referred to its asymptotes xy=k2 (Art. 199) is a particular case of the form ay=B* referred to two tangents and the chord of contact (Arts. 123, 252). Writing the equation xy=(0.x+0.y+k)3, it is evident that the lines x and y are tangents, whose points of contact are at infinity (Art. 154).

254. Again, the equation of a parabola y2=px is also a particular case of ay=62. Writing the equation x (0.x+0.y+p)=y*, the form of the equation shows, not only that the line x touches the curve, its point of contact being the point where x meets y, but also that the line at infinity touches the curve, its point of contact also being on the line y. The same inference may be drawn from the general equation of the parabola

(ax+By)*+ (2gx + 2fy + c) (0.x +0.y + 1) = 0,

which shews that both 2gx + 2fy+c, and the line at infinity are tangents, and that the diameter ax + By joins the points of contact. Thus, then, every parabola has one tangent altogether at an infinite distance. In fact, the equation which determines the direction of the points at infinity on a parabola is a perfect square (Art. 137); the two points of the curve at infinity therefore coincide; and therefore the line at infinity is to be regarded as a tangent (Art. 83).

Ex. The general equation

ax2+2hxy + by2 + 2gx + 2fy + c = 0

=

may be regarded as a particular case of the form (Art. 122) ay kßd. For the first three terms denote two lines a, y passing through the origin, and the last three terms denote the line at infinity ß, together with the line 6, 2gx+2y+c. The form of the equation then shows that the lines a, y meet the curve at infinity, and also that d represents the line joining the finite points in which ay meet the curve.

=

255. In accordance with Art. 253, the equation Skß is to be regarded as a particular case of Saß, and denotes a system of conics passing through the two finite points where ẞ meets S, and also through the two infinitely distant points where S is met by 0.x +0.y+k. Now it is plain that the coefficients of x", of xy, and of y', are the same in S and in S-k3, and therefore (Art. 234) that these equations denote conics similar and similarly placed. We learn, therefore, that two conics similar and similarly placed meet each other in two infinitely distant points, and consequently only in two finite points.

Y

This is also geometrically evident when the curves are hyperbolas; for the asymptotes of similar conics are parallel (Art. 235), that is, they intersect at infinity; but each asymptote intersects its own curve at infinity; consequently the infinitely distant point of intersection of the two parallel asymptotes is also a point common to the two curves. Thus, on the figure, the infinitely distant point of meeting of the lines OX, Ox,

X

and of the lines OY, Oy, are common to the curves. One of their finite points of intersection is shown on the figure, the other is on the opposite branches of the hyperbolas.

If the curves be ellipses, the only difference is that the asymptotes are imaginary instead of being real. The directions of the points at infinity, on two similar ellipses, are determined from the same equation (ax+2hxy + by2 = 0) (Arts. 136, 234), Now, although the roots of this equation are imaginary, yet they are, in both cases, the same imaginary roots, and therefore the curves are to be considered as having two imaginary points at infinity common. In fact, it was observed before, that even when the line a does not meet S in real points, it is to be re

garded as a chord of imaginary intersection of S and S-kaß, and this remains true when the line a is infinitely distant.

If the curves be parabolas, they are both touched by the line at infinity (Art. 254); but the direction of the point of contact, depending only on the first three terms of the equation, is the same for both. Hence, two similar and similarly placed parabolas touch each other at infinity. In short, the two infinitely distant points common to two similar conics are real, imaginary, or coincident, according as the curves are hyperbolas, ellipses, or parabolas.

256. The equation S=k, or S=k(0.x+0.y+1)2 is manifestly a particular case of Ska2, and therefore (Art. 252) denotes a conic having double contact with S, the chord of contact being at infinity. Now S-k differs from S only in the constant term. Not only then are the conics similar and similarly placed, the first three terms being the same, but they are also concentric. For the coordinates of the centre (Art. 140) do not involve C, and therefore two conics whose equations differ only in the absolute term are concentric (see also Art. 81). Hence, two similar and concentric conics are to be regarded as touching each other at two infinitely distant points. In fact, the asymptotes of two such conics are not only parallel but coincident; they have therefore not only two points at infinity common, but also the tangents at those points; that is to say, the curves touch.

If the curves be parabolas, then, since the line at infinity touches both curves, S and S-k have with each other, by Art. 251, a contact at infinity of the third order. Two parabolas whose equations differ only in the constant term will be equal to each other; for the curves y=px, y=p (x+n) are obviously equal, and the equations transformed to any new axes will continue to differ only in the constant term. We have seen, too (Art. 205), that the expression for the parameter of a parabola does not involve the absolute term. The parabolas then, S and S-k are equal, and we learn that two equal and similarly placed parabolas whose axes are coincident may be considered as having with each other a contact of the third order at infinity.

257. All circles are similar curves, the terms of the second degree being the same in all. It follows then, from the last

Articles, that all circles pass through the same two imaginary points at infinity, and on that account can never intersect in more than two finite points, and that concentric circles touch each other in two imaginary points at infinity; and on that account can never intersect in any finite point. It will appear hereafter that a multitude of theorems concerning circles are but particular cases of theorems concerning conics which pass through two fixed points.

258. It is important to notice the form l'a2+m3ß2 = n2y2, which denotes a conic with respect to which a, B, y are the sides of a self-conjugate triangle (Art. 99). For the equation may be written in any of the forms

n2y2 — m2ß2 = l'a2; n2y2 — l'a2 = m2ß2; l2x2 + m2ß* = n2y3. The first form shows that ny+mB, ny - mẞ (which intersect in By) are tangents, and a their chord of contact. Consequently the point By is the pole of a. Similarly from the second form ya is the pole of B. It follows, then, that aß is the pole of y; and this also appears from the third form, which shows that the two imaginary lines la ± mß √(− 1) are tangents whose chord of contact is y. Now these imaginary lines intersect in the real point aß, which is therefore the pole of y; although being within the conic, the tangents through it are imaginary. It appears, in like manner, that

aa2+2haß +bB2 = cy"

denotes a conic, such that aß is the pole of y; for the left-hand side can be resolved into the product of factors representing lines which intersect in aß.

COR. If a2 + m2ß2 = n2y2 denote a circle, its centre must be the intersection of perpendiculars of the triangle aßy. For the perpendicular let fall from any point on its polar must pass through the centre.

258*(a). If x = 0, y=0 be any lines at right angles to each other through a focus, and y the corresponding directrix, the equation of the curve is

x2 + y2 = e2y2,

a particular form of the equation of Art. 258. Its form shows that the focus (xy) is the pole of the directrix y, and that the

* This Article was numbered 279 in the previous editions.

polar of any point on the directrix is perpendicular to the line joining it to the focus (Art. 192); for y, the polar of (xy) is perpendicular tox, but x may be any line drawn through the focus.

The form of the equation shows that the two imaginary lines x2+ y2 are tangents drawn through the focus. Now, since these lines are the same whatever y be, it appears that all conics which have the same focus have two imaginary common tangents passing through this focus. All conics, therefore, which have both foci common, have four imaginary common tangents, and may be considered as conics inscribed in the same quadrilateral. The imaginary tangents through the focus (x2+ y2 = 0) are the same as the lines drawn to the two imaginary points at infinity on any circle (see Art. 257). Hence, we obtain the following general conception of foci: "Through each of the two imaginary points at infinity on any circle draw two tangents to the conic; these tangents will form a quadrilateral, two of whose vertices will be real and the foci of the curve, the other two may be considered as imaginary foci of the curve."

Ex. To find the foci of the conic given by the general equation. We have only to express the condition that x - x' + (y — y′) √(− 1) should touch the curve. Substituting then in the formula of Art. 151, for λ, μ, v respectively, 1, √(− 1), − {x′ + y′ √(− 1)}; and equating separately the real and imaginary parts to cypher, we find that the foci are determined as the intersection of the two loci

C (x2 y2)+2Fy – 2Gx + A−B=0, Cxy - Fx - Gỳ + H = 0,

which denote two equilateral hyperbolas concentric with the given conic. Writing the equations

(Сx — G)2 – (Cy – F)2 – G2 – AC – (F2 – BC) = ▲ (a − b),

(CxG) (Cy- F) = FG - CH = Ah ;

the coordinates of the foci are immediately given by the equations

(Сx - G)2 = ▲ (R + a − b); (Cy — F')2 = §▲ (R + b − a),

where A has the same meaning as at p. 153, and R as at p. 158. If the curve is a parabola, C = 0, and we have to solve two linear equations which give

(F2 + G2) x = FH + } (A − B) G ; (F2 + G2) y = GH + } (B — A) F.

259. We proceed to notice some inferences which follow on interpreting, by the help of Art. 34, the equations we have already used. Thus (see Arts. 122, 123) the equation ay = kß3 implies that the product of the perpendiculars from any point of a conic on two fixed tangents is in a constant ratio to the square of the perpendicular on their chord of contact.

The equation ay=kß8, similarly interpreted, leads to the