Ex. 10. Given four points on a conic, the locus of its centre is a conic through the middle points of the sides of the given quadrilateral. (Ex. 15, Art. 328).

Ex. 11. The locus of the point where parallel chords of a circle are cut in a given ratio is an ellipse having double contact with the circle. (Art. 163).

Given four points on a conic, the locus of the pole of any fixed line is a conic passing through the fourth harmonic to the point in which this line meets each side of the given quadrilateral.

If through a fixed point 0 a line be drawn meeting the conic in A, B, and on it a point P be taken, such that {OABP} may be constant, the locus of P is a conic having double contact with the given conic.

355. We may project several properties relating to foci by the help of the definition of a focus, given p. 239, viz. that if F be a focus, and A, B the two imaginary points in which any circle is met by the line at infinity; then FA, FB are tangents to the conic.

Ex. 1. The locus of the centre of a circle touching two given circles is a hyperbola, having the centres of the given circles for foci.

If a conic be described through two fixed points A, B, and touching two given conics which also pass through those points, the locus of the pole of AB is a conic touching the four lines CA, CB, C'A, C'B, where C, C', are the poles of AB with regard to the two given conics.

In this example we substitute for the word 'circle,' "conic through two fixed points A, B," (Art. 257), and for the word 'centre," "pole of the line AB." (Art. 154).

Ex. 2. Given the focus and two points of a conic section, the intersection of tangents at those points will lie on a fixed line. (Art. 191).

Ex. 3. Given a focus and two tangents to a conic, the locus of the other focus is a right line. (This follows from Art. 189).

Ex. 4. If a triangle circumscribe a parabola, the circle circumscribing the triangle passes through the focus, Cor. 4, Art. 223.

Given two tangents, and two points on a conic, the locus of the intersection of tangents at those points is a right line.

Given two fixed points A, B; two tangents FA, FB passing one through each point, and two other tangents to a conic; the locus of the intersection of the other tangents from A, B, is a right line.

If two triangles circumscribe a conic, their six vertices lie on the same conic.*

For if the focus be F, and the two circular points at infinity A, B, the triangle FAB is a second triangle whose three sides touch the parabola.

Ex. 5. The locus of the centre of a circle passing through a fixed point, and touching a fixed line, is a parabola of which the fixed point is the focus.

Given one tangent, and three points on a conic, the locus of the intersection of tangents at any two of these points is a conic inscribed in the triangle formed by those points.

*This is easily proved directly. Take a side of each triangle and, by the anharmonic property of the tangents of a conic, these lines are cut homographically by the other four sides; whence it may easily be seen that the pencils joining the opposite vertices of each triangle to the other four are homographic:

Ex. 6. Given four tangents to a conic, the locus of the centre is the line joining the middle points of the diagonals of the quadrilateral.

Given four tangents to a conic, the locus of the pole of any line is the line joining the fourth harmonics of the points where the given line meets the diagonals of the quadrilateral.

It follows from our definition of a focus, that if two conics have the same focus, this point will be an intersection of common tangents to them, and will possess the properties mentioned at the end of Art. 264. Also, that if two conics have the same focus and directrix, they may be considered as two conics having double contact with each other, and may be projected into concentric circles.

356. Since angles which are constant in any figure will in general not be constant in the projection of that figure, we proceed to show what property of a projected figure may be inferred when any property relating to the magnitude of angles is given; and we commence with the case of the right angle.

Let the equations of two lines at right angles to each other be x=0, y = 0, then the equation which determines the direction of the points at infinity on any circle is x2 + y2 = 0, or

x+y√-1=0, x-y-1=0.

Hence (Art. 57) these four lines form a harmonic pencil. Hence, given four points A, B, C, D, of a line cut harmonically, where A, B may be real or imaginary, if these points be transferred by a real or imaginary projection, so that A, B may become the two imaginary points at infinity on any circle, then any lines through C, D will be projected into lines at right angles to each other. Conversely, any two lines at right angles to each other will be projected into lines which cut harmonically the line joining the two fixed points which are the projections of the imaginary points at infinity on a circle.

Ex. 1. The tangent to a circle is at right angles to the radius.

Any chord of a conie is cut harmonically by any tangent, and by the line joining the point of contact of that tangent to the pole of the given chord. (Art. 146).

For the chord of the conic is supposed to be the projection of the line at infinity in the plane of the circle; the points where the chord meets the conic will be the projections of the imaginary points at infinity on the circle; and the pole of the chord will be the projection of the centre of the circle.

Any right line through a point, the line joining its pole to that point, and the two tangents from the point, form a harmonic pencil. (Art. 146).

Ex. 2. Any right line drawn through the focus of a conic is at right angles to the line joining its pole to the focus. (Art. 192). It is evident that the first of these properties is only a particular case of the TT.

second, if we recollect that the tangents from the focus are the lines joining the focus to the two imaginary points on any circle.

Ex. 3. Let us apply Ex. 6 of the last Article to determine the locus of the pole of a given line with regard to a system of confocal conics. Being given the two foci, we are given a quadrilateral circumscribing the conic (Art. 258a); one of the diagonals of this quadrilateral is the line joining the foci, therefore (Ex. 6) one point on the locus is the fourth harmonic to the point where the given line cuts the distance between the foci. Again, another diagonal is the line at infinity, and since the extremities of this diagonal are the points at infinity on a circle, therefore by the present Article the locus is perpendicular to the given line. The locus is, therefore, completely determined.

Ex. 4. Two confocal conics cut each other at right angles.

If two conics be inscribed in the same quadrilateral, the two tangents at any of their points of intersection cut any diagonal of the circumscribing quadrilateral harmonically.

The last theorem is a case of the reciprocal of Ex. 1, Art. 345. Ex. 5. The locus of the intersection of two tangents to a central conic, which cut at right angles, is a circle.

The locus of the intersection of tangents to a conic, which divide harmonically a given finite right line AB, is a conic through A, B.

The last theorem may, by Art. 146, be stated otherwise thus: "The locus of a point O, such that the line joining O to the pole of A0 may pass through B, is a conic through A, B ;" and the truth of it is evident directly, by taking four positions of the line, when we see, by Ex. 2, Art. 297, that the anharmonic ratio of four lines AO is equal to that of four corresponding lines BO.

Ex. 6. The locus of the intersection of tangents to a parabola, which cut at right angles, is the directrix.

Ex. 7. The circle circumscribing a triangle self-conjugate with regard to an equilateral hyperbola passes through the centre of the curve. (Ex. 5, Art. 228).

If in the last example AB touch the given conic, the locus of O will be the line joining the points of contact of tangents from A, B.

If two triangles are both self-conjugate with regard to a conic, their six vertices lie on a conic.

The fact that the asymptotes of an equilateral hyperbola are at right angles may be stated, by this Article, that the line at infinity cuts the curve in two points which are harmonically conjugate with respect to A, B, the imaginary circular points at infinity. And since the centre C is the pole of AB, the triangle CAB is self-conjugate with regard to the equilateral hyperbola. It follows, by reciprocation, that the six sides of two self-conjugate triangles touch the same conic.

Ex. 8. If from any point on a conic If a harmonic pencil be drawn through two lines at right angles to each other be any point on a conic, two legs of which drawn, the chord joining their extremities are fixed, the chord joining the extremities passes through a fixed point. (Ex. 2, of the other legs will pass through a fixed Art. 181). point. In other words, given two points a, c on a conic, and {abcd} a harmonic ratio, bd will pass through a fixed point, namely, the intersection of tangents at a, c. But the truth of this may be seen directly: for let the line ac meet bd in K, then, since {a.abcd} is a harmonic pencil, the tangent at a cuts bd in the fourth harmonic to K: but so likewise must the tangent at c, therefore these tangents meet bd in the same point. As a particular case of this theorem we have the following: "Through a fixed

point on a conic two lines are drawn, making equal angles with a fixed line, the chord joining their extremities will pass through a fixed point."

357. A system of pairs of right lines drawn through a point, so that the lines of each pair make equal angles with a fixed line, cuts the line at infinity in a system of points in involution, of which the two points at infinity on any circle form one pair of conjugate points. For they evidently cut any right line in a system of points in involution, the foci of which are the points where the line is met by the given internal and external bisector of every pair of right lines. The two points at infinity just mentioned belong to the system, since they also are cut harmonically by these bisectors.

The tangents from any point to a system of confocal conics make equal angles with two fixed lines. (Art. 189).

The tangents from any point to a system of conics inscribed in the same quadrilateral cut any diagonal of that quadrilateral in a system of points in involution of which the two extremities of that diagonal are a pair of conjugate points. (Art. 344).

358. Two lines which contain a constant angle cut the line joining the two points at infinity on a circle, so that the anharmonic ratio of the four points is constant.

For the equation of two lines containing an angle being x = 0, y = 0, the direction of the points at infinity on any circle is determined by the equation

x2 + y2+2xy cos 0 = 0;

and, separating this equation into factors, we see, by Art. 57, that the anharmonic ratio of the four lines is constant if Ø be constant.

Ex. 1. "The angle contained in the same segment of a circle is constant." We see, by the present Article, that this is the form assumed by the anharmonic property of four points on a circle when two of them are at an infinite distance.

Ex. 2. The envelope of a chord of a Iconic which subtends a constant angle at the focus is another conic having the same focus and the same directrix.

Ex. 3. The locus of the intersection of tangents to a parabola which cut at a given angle is a hyperbola having the same focus and the same directrix.

If tangents through any point 0 meet the conic in T, T', and there be taken on the conic two points A, B, such that {0.ATBT'} is constant, the envelope of AB is a conic touching the given conic in the points T, T'.

If a finite line AB, touching a conic be cut by two tangents in a given anharmonic ratio, the locus of their intersection is a conic touching the given conic at the points of contact of tangents from A, B.

Ex. 4. If from the focus of a conic a line be drawn making a given angle with any tangent, the locus of the point where it meets it is a circle.

If a variable tangent to a conic meet two fixed tangents in T, T', and a fixed line in M, and there be taken on it a point P, such that {PTMT'} may be constant, the locus of P is a conic passing through the points where the fixed tangents meet the fixed line.

A particular case of this theorem is: "The locus of the point where the intercept of a variable tangent between two fixed tangents is cut in a given ratio is a hyperbola whose asymptotes are parallel to the fixed tangents."

Ex. 5. If from a fixed point O, OP be drawn to a given circle, and TP be drawn making the angle TPO constant, the envelope of TP is a conic having O for its focus.

Given the anharmonic ratio of a pencil three of whose legs pass through fixed points, and whose vertex moves along a given conic, passing through two of the points, the envelope of the fourth leg is a conic touching the lines joining these two to the third fixed point.

A particular case of this is: "If two fixed points A, B on a conic be joined to a variable point P, and the intercept made by the joining chords on a fixed line be cut in a given ratio at M, the envelope of PM is a conic touching parallels through A and B to the fixed line.

Ex. 6. If from a fixed point 0, OP be drawn to a given right line, and the angle TPO be constant, the envelope of TP is a parabola having O for its focus.

Given the anharmonic ratio of a pencil, three of whose legs pass through fixed points, and whose vertex moves along a fixed line, the envelope of the fourth leg is a conic touching the three sides of the triangle formed by the given points.

359. We have now explained the geometric method by which, from the properties of one figure, may be derived those of another figure which corresponds to it (not as in Chap. XV., so that the points of one figure answer to the tangents of the other, but) so that the points of one answer to the points of the other, and the tangents of one to the tangents of the other. All this might be placed on a purely analytical basis. If any curve be represented by an equation in trilinear coordinates, referred to a triangle whose sides are a, b, c, and if we interpret this equation with regard to a different triangle of reference whose sides are a', b', c', we get a new curve of the same degree as the first ;* and the same equations which establish any property of the first curve will, when differently interpreted, establish

It is easy to see that the equation of the new curve referred to the old triangle is got by substituting in the given equation for a, ß, y; la+mẞ+ny, l'a+m'ß+n'y, l''a+m"ß+n"y, where la + mẞ+ ny represents the line which is to correspond to a, &c. For fuller information on this method of transformation see Higher Plane Curves, Chap. VIII,