the lines L, N to be the axes, we obtain the known form of the equation of a conic referred to its asymptotes ay = m2 (Art. 204).

In like manner, the equation IN = M2 (where 7 is a constant) denotes a conic to which N is one tangent, and 7, the line at infinity, is another. In this equation the highest terms form the perfect square M2, and therefore the curve is a parabola. Conversely, 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. 136); the two points of the curve at infinity therefore coincide; and therefore the line at infinity is to be regarded as a tangent (Art. 81). And the form of the equation of the parabola px = y2 denotes that the line at infinity p is one tangent, the line x another, and that the diameter y is the line joining their points of contact. So, in general, the equation

(ax + by)2 + Dx + Ey + F = 0

denotes a parabola to which Dx + Ey + F = 0 is a tangent, and ax + by O the diameter through the point of contact.

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256. In like manner, it may be inferred from Art. 253 that the equations S = 0, S + 7M = 0 (where is a constant), denote two conics intersecting each other in the two finite points where M meets either, and also in the two infinitely distant points where the line at infinity 7 meets either. Now, it is plain that the coefficients of x2, xy, and y' are the same in the two equations S = 0, S + 7M = 0; and therefore (Art. 240) that these equations denote two conics similar and similarly placed. We learn, therefore, that two conics similar and similarly placed can cut each other only in two finite points; and that this is because they also cut each other in two real, coincident, or imaginary points at infinity.

257. We may arrive geometrically at the same conclusion.

First. If the curves be hyperbolæ.

The asymptotes of similar hyperbolæ are parallel (Art. 241), that is, they intersect each other at infinity; but each asymptote intersects its own curve at in

Y

X

finity; hence we infer that similar and similarly placed hyperbolæ intersect each other in the two points at infinity, where each is intersected by its own asymptotes (see the figure, where the two hyperbolæ evidently tend to intersect at the two points at infinity, where OX meets ox, and OY meets oy).

Secondly. If the curves be ellipses. Ellipses only differ from hyperbole in having imaginary instead of real asymptotes. The directions of the points at infinity on either of two similar ellipses are determined from the same equation (Ax2 + Bxy + Cy2 = 0) (Arts. 134 and 240). Now, although the roots of this equation are in both cases imaginary, yet they are in both cases the same imaginary roots; we infer, therefore, that two similar ellipses pass through the same two imaginary points at infinity.

Thirdly. If the curves be parabola. They are both touched by the line at infinity (Art. 255). The direction of the point of contact at infinity is the same as that of the diameters (Art. 140), and is therefore the same for two similarly placed parabolæ (Art. 242). Hence two similarly placed parabolæ touch each other at infinity.

258. It may be inferred in precisely the same way, from Art. 254, that the equation S+7= 0, where I is constant, denotes a conic touching the conic S in two points at infinity. Now if the equations of two conics only differ in the constant terms, since the co-ordinates of the centre do not contain F (Art. 138), the conics must have the same centre; and since the first three terms are the same in both, the conics are similar; hence the conics S and S + 72 are similar and concentric. We learn then that similar and concentric conics are to be regarded as touching each other at two points at an infinite distance. This is otherwise evident, since we have proved in the last Article that the curves pass through the same points at infinity; and since they have the same, real or imaginary, asymptotes, they have also the same tangents at those points.

If the curves be parabolæ, then since the line at infinity touches both, by Art. 254, the conics S and S+ 2 have with each other a contact of the third order at infinity. Two parabola whose equations only differ in the constant term will be

equal to each other; for the parabolæ y2 = px, and y2 = p (x + n), are obviously equal, and if the origin be transferred to any other point the equations will continue to differ only in the constant term. We have seen too (Art. 213) that the expression for the parameter of a parabola does not involve the absolute term. The parabolæ, then, S and S + l2, are equal to each other, and we learn that two equal and similarly placed parabolæ may be considered as having with each other a contact of the third order at infinity.

259. Since all circles are similar curves, it follows, as a particular case of the last Articles, that all circles pass through the same two imaginary points at infinity, and that concentric circles touch each other in two imaginary points at infinity. Thus we see the reason why two circles cannot cut each other in more than two finite points, and why two concentric circles do not meet in any finite point, although two curves of the second degree in general intersect in four points. We shall also show that the theorems established (p. 103, &c.), concerning circles which pass through the same two points, are only particular cases of more general theorems concerning conic sections which pass through the same four points.

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260. We proceed to notice some inferences which follow immediately on interpreting the preceding equations by the help of Art. 27. Thus the equation ay kß 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 important theorem: The product of the perpendiculars let fall `from any point of a conic on two opposite sides of an inscribed quadrilateral is in a constant ratio to the product of the perpendiculars let fall on the other two sides.

From this property we at once infer, that the anharmonic ratio of a pencil, whose sides pass through four fixed points of a conic, and whose vertex is any variable point of it, is constant.

For the perpendicular

but the right-hand member of this equation is constant, while the left-hand member is the anharmonic ratio of the pencil OA, OB, OC, OD.

The consequences of this theorem are so numerous and important, that we shall devote a section of the next chapter to develop them more fully.

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261. If S=0 be the equation to a circle, then (Art. 88) S is the square of the tangent from any point xy to the circle; hence S-kaẞ0 (the equation of a conic whose chords of intersection with the circle are a and B) expresses that the locus of a point, such that the square of the tangent from it to a fixed circle is in a constant ratio to the product of its distances from two fixed lines, is a conic passing through the four points in which the fixed lines intersect the circle.

This theorem is equally true whatever be the magnitude of the circle, and whether the right lines meet the circle in real or imaginary points; thus, for example, if the circle be infinitely small, the locus of a point, the square of whose distance from a fixed point is in a constant ratio to the product of its distances from two fixed lines, is a conic section; and the fixed lines may be considered as chords of imaginary intersection of the conic with an infinitely small circle whose centre is the fixed point.

262. Similar inferences can be drawn from the equation Ska2 = 0, where S is a circle. We learn that the locus of a point, such that the tangent from it to a fixed circle is in a constant ratio to its distance from a fixed line, is a conic touching the circle

at the two points where the fixed line meets it; or, conversely, that if a circle have double contact with a conic, the tangent drawn to the circle from any point on the conic is in a constant ratio to the perpendicular from the point on the chord of contact.

In the particular case where the circle is infinitely small, we obtain the fundamental property of the focus and directrix, and we infer that the focus of any conic may be considered as an infinitely small circle, touching the conic in two imaginary points situated on the directrix.

263. In general, if in the equation of any conic the co-ordinates of any point be substituted, the result will be proportional to the rectangle under the segments of a chord drawn through the point parallel to a given line.*

For (Art. 151) this rectangle

=

F'

A cos20+B cos sin 0 + C sin20'

where, by Art. 129, F' is the result of substituting in the equation the co-ordinates of the point; if, therefore, the angle be constant, this rectangle will be proportional to F'. Hence, we may extend the last-proved theorems to the case where S is any conic. For example: "If two conics have double contact, the square of the perpendicular from any point of one upon the chord of contact, is in a constant ratio to the rectangle under the segments of that perpendicular made by the other;" or, in general, "If a line parallel to a given one meets two conics in the points P, Q, p, q, and we take on it a point O, such that the rectangle OP. OQ may be to Op. Oq in a constant ratio, the locus of O is a conic through the points of intersection of the given conics."

264. If two conics have each double contact with a third, their chords of contact with the third conic, and a pair of their chords of intersection with each other, will all pass through the same point, and will form an harmonic pencil.

Let the equation of the third conic be S = 0, and those of the other two conics,