and we can then determine A, H, B from the equations

Ab+ Ba-2Hh=0, Ab' + Ba' - 2 Hh'=0.

We see, as in Art. 333, that any other pair of points in involution with the two given pairs may be represented by an equation of the form

(ax3 + 2hx + b) +λ (a'x2 + 2h'x + b′) = 0,

since, when A, H, B are determined so as to satisfy the two equations written above, they must also satisfy

A (b + λb') + B (a + λa') − 2H (h + λh') = 0.*

The actual values of A, B, H, found by solving these equations, are 2 (ah' — a'h), 2 (hb′ — h′b), ab′ — ab. Consequently the foci of the system determined by the given pairs of points, are given by the equation

(ah' – a′h) x2 + (ab′ – a′b) x + (hb′ – h’b) = 0.

This may be otherwise written if we make the equations homogeneous by introducing a new variable y, and write U=ax2+2hxy + by*, V=a'x*+2h'xy + b'y*.

The equation which determines the foci is then

The foci of a system given by two pairs of points a, a′; b, b′ may be also found as follows, from the consideration that {afba} {a'fb'a), or

whence

=

af.ba aƒ.b'a
af.ba

=

af.b'a'

aƒa: a'ƒ“ :: ab.ab' : a'b.a'b';

orf is the point where aa' is cut either internally or externally in a certain given ratio.

343. The relation connecting six points in involution is of the class noticed in Art. 313, and is such that the same relations

* It easily follows from this, that the condition that three pairs of points ax2+2hx + b, a'x2 + 2h'x + b', a′′x2 + 2h′′x+b" should belong to a system in involution, is the vanishing of the determinant

will subsist between the sines of the angles subtended by them at any point as subsist between the segments of the lines themselves. Consequently, if a pencil be drawn from any point to six points in involution, any transversal cuts this pencil in six points in involution. Again, the reciprocal of six points in involution is a pencil in involution.

The greater part of the equations already found apply equally to lines drawn through a point. Thus, any pair of lines a-μẞ, a-μ'ß belong to a system in involution, if

Aμμ' + ♬ (μ + μ') + B = 0;

and if we are given two pairs of lines

U=aa* + 2haß +bß2, V=a'a2+2h′aß +b′ß2,

they determine a pencil in involution whose focal lines are (ah' — a’h) a2 + (ab′ — a′b) aß + (hb′ — h’b) ß2 = 0,

344. A system of conics passing through four fixed points meets any transversal in a system of points in involution.

For, if S, S' be any two conics through the points, S+λS' will denote any other; and if, taking the transversal for axis of x and making y=0 in the equations, we get ax2 + 2gx + c, and a'x2+ 2g'x+c to determine the points in which the transversal meets S and S', it will meet S+λS' in

ax2+2gx+c+λ (a′x2 + 2g′x+c′),

a pair (Art. 342) in involution with the two former pair. This may also be proved

geometrically as follows:

By the anharmonic properties of conics,

{a.Adb A'}={c. AdbA'}:

but if we observe the points

in which these pencils meet

AA', we get {ACBA'} = {AB′&A'} = {A′C′B'A}.

Consequently the points AA' belong to the system in involution determined by BB, CC', the pairs of points in which

the transversal meets the sides of the quadrilateral joining the given points.

Reciprocating the theorem of this article we learn that, the pairs of tangents drawn from any point to a system of conics touching four fixed lines, form a system in involution.

345. Since the diagonals ac, bd may be considered as a conic through the four points, it follows, as a particular case of the last Article, that any transversal cuts the four sides and the diagonals of a quadrilateral in points BB, CC, DD', which are in involution. This property enables us, being given two pairs of points BB, DD' of a system in involution, to construct the point conjugate to any other C. For take any point at random, a; join aB, aD, a C; construct any triangle bcd, whose vertices rest on these three lines, and two of whose sides pass through B'D', then the remaining side will pass through C', the point conjugate to C. The point a may be taken at infinity, and the lines aB, aD, a C will then be parallel to each other. If the point C be at infinity the same method will give us the centre of the system. The simplest construction for this case is,-"Through B, D, draw any pair of parallel lines Bb, Dc; and through B, D', a different pair of parallels D'b, B'c; then be will pass through the centre of the system.

Ex. 1. If three conics circumscribe the same quadrilateral, the common tangent to any two is cut harmonically by the third. For the points of contact of this tangent are the foci of the system in involution.

Ex. 2. If through the intersection of the common chords of two conics we draw a tangent to one of them, this line will be cut harmonically by the other. For in this case the points D and D' in the last figure coincide, and will therefore be a focus. Ex. 3. If two conics have double contact with each other, or if they have a contact of the third order, any tangent to the one is cut harmonically at the points where it meets the other, and where it meets the chord of contact. For in this case the common chords coincide, and the point where any transversal meets the chord of contact is a focus.

Ex. 4. To describe a conic through four points a, b, c, d, to touch a given right line. The point of contact must be one of the foci of the system BB', CC", &c., and these points can be determined by Art. 342. This problem, therefore, admits of two solutions.

Ex. 5. If a parallel to an asymptote meet the curve in C, and any inscribed quadrilateral in points abcd; Ca. Cc Cb. Cd. For C is the centre of the system.

Ex. 6. Solve the examples, Art. 326, as cases of involution.

In Ex. 1, K is a focus: in Ex. 2, T is also a focus: in Ex. 3, T is a centre, &c. Ex. 7. The intercepts on any line between a hyperbola and its asymptotes are equal. For in this case one focus of the system is at infinity (Cor., Art. 341).

346. If there be a system of conics having a common self-conjugate triangle, any line passing through one of the vertices of this triangle is cut by the system in involution.

For, if in aa2 + bẞ" + cy3 we write a = kß, we get (ak2 + b) B2 + cy*,

a pair of points evidently always harmonically conjugate with the two points where the line meets B and y. Thus, then, in particular, a system of conics touching the four sides of a fixed. quadrilateral cuts in involution any transversal which passes through one of the intersections of diagonals of the quadrilateral (Ex. 3, Art. 146). The points in which the transversal meets diagonals are the foci of the system, and the points where it meets opposite sides of the quadrilateral are conjugate points of the system.

Ex. 1. If two conics U, V touch their common tangents A, B, C, D in the points a, b, c, d; a', b', c', d'; a conic S through the points a, b, c, and touching D at d', will have for its second chord of intersection with V, the line joining the intersections of A with bc, B with ca, C with ab.

Let V meet ab in a, ß, then, by this article, since ab passes through an intersection of diagonals of ABCD (Ex. 2, Art. 263), a, b ; a, ẞ belong to a system in involution, of which the points where ab meets C and D are conjugate points. But (Art. 345) the common chords of S and V meet ab in points belonging to this same system in involution, determined by the points a, b; a, ß, in which S and V meet the line ab. If then one of the common chords be D, the other must pass through the intersection of C with ab.

Ex. 2. If in a triangle there be inscribed an ellipse touching the sides at their middle points a, b, c, and also a circle touching at the points a', b', c', and if the fourth common tangent D to the ellipse and circle touch the circle at d', then the circle described through the middle points touches the inscribed circle at d. By Ex. 1, a conic described through a, b, c, will touch the circle at d', if it also pass through the points where the circle is met by the line joining the intersections of A, bc; B, ca; C, ab. But this line is in this case the line at infinity. The touching conic is therefore a circle. Sir W. R. Hamilton has thus deduced Feuerbach's theorem (p. 127) as a particular case of Ex. 1.

The point d' and the line D can be constructed without drawing the ellipse. For since the diagonals of an inscribed, and of the corresponding circumscribing quadrilateral meet in a point, the lines ab, cd; a'b', c'd', and the lines joining AD, BC; AC, BD all intersect in the same point. If then a, ẞ, y be the vertices of the triangle formed by the intersections of bc, b'c'; ca, c'a'; ab, a'b'; the lines joining a'a, b'ß, c'y meet in d. In other words, the triangle aßy is homologous with abc, a'b'c', the centres of homology being the points d, d'. In like manner, the triangle aßy is also homologous with ABC, the axis of homology being the line D.

SS.

CHAPTER XVII.

THE METHOD OF PROJECTION.*

347. WE have already several times had occasion to point out to the reader the advantage gained by taking notice of the number of particular theorems often included under one general enunciation, but we now propose to lay before him a short sketch of a method which renders us a still more important service, and which enables us to tell when from a particular given theorem we can safely infer the general one under which it is contained.

If all the points of any figure be joined to any fixed point in space (0), the joining lines will form a cone, of which the point is called the vertex, and the section of this cone, by any plane, will form a figure which is called the projection of the given figure. The plane by which the cone is cut is called the plane of projection.

To any point of one figure will correspond a point in the other. For, if any point A be joined to the vertex O, the point a, in which the joining line OA is cut by any plane, will be the projection on that plane of the given point A.

A right line will always be projected into a right line.

For, if all the points of the right line be joined to the vertex, the joining lines will form a plane, and this plane will be intersected by any plane of projection in a right line.

Hence, if any number of points in one figure lie in a right line, so will also the corresponding points on the projection; and if any number of lines in one figure pass through a point, so will also the corresponding lines on the projection.

*This method is the invention of M. Poncelet. See his Traité des Propriétés Projectives, published in the year 1822, a work which I believe may be regarded as the foundation of the Modern Geometry. In it were taught the principles, that theorems concerning infinitely distant points may be extended to finite points on a right line; that theorems concerning systems of circles may be extended to conics having two points common; and that theorems concerning imaginary points and lines may be extended to real points and lines.