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.

=

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 @ + 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,

Now, on subtracting these equations, we find for the equation of the chords of intersection,

The chords of intersection, therefore (L M = 0, L + M = 0), pass through the intersection of the chords of contact (L and M), and form an harmonic pencil with them (Art. 55).

It is important that the student should acquire the habit of taking notice of the number of particular theorems often included under one general enunciation; thus, for example, the present theorem holds good, and is proved, in like manner, if the conic S reduce to two right lines; hence, the chords of contact of two conics with their common tangents pass through the intersection of their common chords.

Again, if S be any conic, while S+L and S+ M2 both reduce to pairs of right lines, these right lines will then form a circumscribing quadrilateral, and the chords of intersection (L2 – M2) will be the diagonals of that quadrilateral, while the chords of contact (L and M) obviously are the diagonals of the inscribed quadrilateral formed by joining the points of contact. Hence, the diagonals of any inscribed, and of the corresponding circumscribed quadrilateral, pass through the same point, and form an harmonic pencil.

=

The theorem of this Article may also be stated thus: If a conic section pass through two given points, and have double contact with a given conic, the chord of contact passes through a fixed point. For, suppose any conic (S+ L2 0) through the two given points to be fixed, then the intersection of its chord of contact (L), with the line joining the given points, determines a point through which, by the present Article, any other chord of contact must pass.

In like manner: Given two tangents and two points on a conic section; the chord of contact will pass through a fixed point on the line joining the two given points.

265. If three conics have each double contact with a fourth, their six chords of intersection will pass three by three through the same points, thus forming the sides and diagonals of a quadrilateral.

As in the last Article, we may deduce hence many particular theorems, by supposing one or more of the conics to break up into right lines.

Thus, for example, if S break up into right lines, it represents two common tangents to S + M2, S + N2; and if L denote any right line through the intersection of those common tangents, then S + L2 also breaks up into right lines, and represents any two right lines passing through the intersection of the common tangents. Hence, if through the intersection of the common tangents of two conics we draw any pair of right lines, the chords of each conic joining the extremities of those lines will meet on one of the common chords of the conics. This is the extension of Art. 121. Or, again, tangents at the extremities of either of these right lines will meet on one of the common chords.

266. If SL, S + M2, S + N2, all break up into pairs of right lines, they will form a hexagon circumscribing S, the chords of intersection will be diagonals of that hexagon, and the proposition of this Article becomes Brianchon's theorem: "The three opposite diagonals of every hexagon circumscribing a conic intersect in a point."

By the opposite diagonals we mean (if the sides of the hexagon be numbered 1, 2, 3, 4, 5, 6) the lines joining (1, 2) to (4, 5), (2, 3) to (5, 6), and (3, 4) to (6, 1); and by changing the order in which we take the sides, we may consider the same lines as forming a number (sixty) of different hexagons, for each of which the present theorem is true.

By supposing two sides of the hexagon to be indefinitely near, we obtain from this theorem a very simple construction for the solution of the problem,-"Given five tangents, to find the point

of contact of any of them,"-since any tangent is intersected by a consecutive tangent at its point of contact (p. 130).

267. If three conic sections have one chord common to all, their three other common chords will pass through the same point. Let the equation of one be S 0, and of the common chord L = 0, then the equations of the other two are of the form

S + LM = 0,

=

which must have, for their intersection with each other,

L(M - N) = 0;

but M - N is a line passing through the point (MN).

According to the remark in Art. 259, this is only an extension of the theorem (Art. 113), that the radical axes of three circles meet in a point. For three circles have one chord (the line at infinity) common to all, and the radical axes are their other common chords.

The theorem of Art. 265 may be considered as a still further extension of the same theorem, and three conics which have each double contact with a fourth may be considered as having four radical centres, through each of which pass three of their common chords.

The theorem of this Article may, as in Art. 113, be otherwise enunciated: Given four points on a conic section, its chord of intersection with a fixed conic passing through two of these points will pass through a fixed point.

PD

A number of particular inferences may also be drawn from the theorem of the present Article, by supposing one or more of the conics to break up into two right lines. Thus, for example, if one of the conics break up into the pair of lines OA, OB, we obtain the theorem: "If through one of the points of intersection

A

of two conics we draw any line meeting the conics in the points P, p, and through any other point of intersection B a line meeting the conics in the points Q, q, then the lines PQ, pq, will meet on CD, the other chord of intersection." Next let the points A, B coincide, then the two conics will touch at A, and we learn