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. 279); 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 Ex. 5. The locus of the intersection of two tangents to a central conic, which cut at right angles, a circle.

of Ex. 1, p. 300.

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 0, such that the line joining 0 to the pole of 40 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, p. 260, 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. (p. 204).

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 two lines at right angles to each other be drawn, the chord joining their extremities passes through a fixed point. (p. 170).

If a harmonic pencil be drawn through any point on a conic, two legs of which are fixed, the chord joining the extremities of the other legs will pass through a fixed point.

In other words, given two points, a, c, on a conic, and {abcd} a harmonic ratio, ¿d 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 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 0, OP be drawn to a given circle, and the angle TPO be 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 0 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 co-ordinates, 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. VI.

a corresponding property of the second. In this manner a right line in one system always-corresponds to a right line in the other, except in the case of the equation aa+bB+ cy=0, which in the one system represents an infinitely distant line, in the other a finite line. And, in like manner, a'a+b'ß + c'y, which represents an infinitely distant line in the second system represents a finite line in the first system. In working with trilinear co-ordinates the reader can hardly have failed to take notice, how the method itself teaches him to generalize all theorems in which the line at infinity is concerned. Thus (see p. 243) if it be required to find the locus of the centre, of a conic, when four points or four tangents are given, this is done by finding the locus of the pole of the line at infinity aa+b+cy, and the very same process gives the locus under the same conditions of the pole of any line λα + β + νγ.

We saw (Art. 59) that the anharmonic ratio of a pencil P−kP', P−lP', &c. depends only on the constants k, l, and is not changed if Pand P' are supposed to represent different right lines. We can infer then that in the method of transformation which we are describing, to a pencil of four lines in the one system answers in the other system a pencil having the same anharmonic ratio; and that to four points on a line correspond four points whose anharmonic ratio is the same.

An equation, S=0, which represents a circle in the one system will, in general, not represent a circle in the other. But since any other circle in the first system is represented by an equation of the form

S+ (aa+bB+cy) (λa + μß + vy) = 0,

all curves of the second system answering to circles in the first will have common the two points common to S and aa+bB+cy.

360. In this way we are led, on purely analytical grounds, to the most important principles, on the discovery and application of which the merit of Poncelet's great work consists. The principle of continuity (in virtue of which properties of a figure in which certain points and lines are real, are asserted to be true even when some of these points and lines are imaginary),

SS

is more easily established on analytical than on purely geometrical grounds. In fact, the processes of analysis take no account of the distinction between real and imaginary, so important in pure geometry. The processes for example by which, in Chap. XIV. we obtained the properties of systems of conics represented by equations of forms S-kaß or S-ka2 are unaffected, whether we suppose a and B to meet S in real or imaginary points. And though from any given property of a system of circles, we can obtain, by a real projection, only a property of a system of conics having two imaginary points common, yet it is plainly impossible to prove such a property by general equations without proving it at the same time for conics having two real points common. The analytical method of transformation, described in the last article, is equally applicable if we wish real points in one figure to correspond to imaginary points on the other. Thus, for example, a* + ß2 = y2 denotes a curve met by y in imaginary points; but if we substitute for a, ß; P±Q √ (− 1), and for y, R, where P, Q, R denote right lines, we get a curve met in real points by R the line corresponding to y.

The chief difference in the application of the method of projections, considered geometrically and considered algebraically, is that the geometric method would lead us to prove a theorem first for the circle or some other simple state of the figure, and then infer a general theorem by projection. The algebraic method finds it as easy to prove the general theorem as the simpler one, and would lead us to prove the general theorem first, and afterwards infer the other as a particular

case.

THEORY OF THE SECTIONS OF A CONE.

361. The sections of a cone by parallel planes are similar. Let the line joining the vertex O to any fixed point A in one plane, meet the other in the point a; and let radii vectores be drawn from A, a, to any other two corresponding points B, b. Then, from the similar triangles OAB, Oab, AB is to ab in the constant ratio OA: Oa; and since every radius vector of the one curve is parallel and in a constant ratio to the corresponding radius vector of the other, the two curves are similar (Art. 233).