it follows that [pqrs] = [p'q'r's'], or p, p, q, q', r, r'... are a system of points in involution. Hence, any system of points in involution projects into a system in involution.

If P coincides with P', p will coincide with p', or the foci of one system project into the foci of the other. We may observe that the centre of one system will not, in general, project into the centre of the other.

26. Let a system of circles be described through two given points A, A', and let any circle of the system cut a given straight line in P, P'. Produce AA' to meet the given straight line in O. Then

or OP. OP' is constant for all circles passing through A, A'. Hence, the system of points in which a system of circles, passing through two given points, cut a given straight line, are in involution. Project the system of circles into a system of conics, passing through four given points, and we learn that "a system of conics, passing through four given points, cut any straight line in a system of points in involution."

Of this system of conics, one can be drawn so that one of its points of intersection with the given straight line shall be at an infinite distance,-in other words, so that one of its asymptotes shall be parallel to the given straight line. The other point, in which this conic cuts the given straight line, will be the centre of the system.

Again (see Art. 3, Chap. IX., infra), two conics can be described, passing through the four given points, and touching the given straight line. The two points of contact of these conics will be the foci of the system of points in involution.

By reciprocating these propositions, we obtain analogous properties of the system of conics, inscribed in a given quadrilateral, whence, by projection, may be obtained those of a system of confocal conics.

27. When the vertex of the cone, used for purposes of projection, is infinitely distant, so that the cone itself be

comes a cylinder, the projection is said to be orthogonal. In this mode of projection, the line at infinity remains at an infinite distance, and any two parallel lines will therefore project into parallel lines. Also any area will bear to its projection a constant ratio; and the mutual distances of any three points in the same straight line will bear to one another the same ratios as the mutual distances of their projections. Two perpendicular diameters of a circle will, since each is parallel to the tangent at the extremity of the other, project into two conjugate diameters of an ellipse. By this method, many properties of conic sections, more especially those relating to conjugate diameters, may be readily deduced from those of the circle.

EXAMPLES.

1. If XYZ be a triangle which moves in such a manner that its side YZ always passes through a fixed point P, ZX through Q, XY through R, and if the locus of Y be a fixed conic passing through R and P, that of Z a fixed conic passing through P and Q, prove that the locus of X will be a fixed conic passing through Q, R, and through the other three points of intersection of the two given conics.

2. If two tangents be drawn to a conic so that the points in which they cut a given straight line form, with two fixed points on the straight line, a harmonic range, prove that the locus of their point of intersection will be a conic passing through the two given points.

3. A system of conics is described touching four given straight lines; prove that the locus of the pole of any fifth given straight line with respect to any conic of the system is a straight line.

If the fifth straight line be projected to infinity so that the points where it intersects two of the other given straight lines be projected into the circular points, what does this theorem become?

4. A system of conics is described about a given quadrangle; prove that the locus of the pole of any given straight line, with respect to any conic of the system, is a conic passing through the vertices of the quadrangle.

5. A system of conics is described touching the sides of a given triangle, and from a given point a pair of tangents is drawn to each conic of the system. Prove that, if the locus of one of the points of contact be a straight line, that of the other will be a conic circumscribed about the given triangle.

6. The tangent at any point P of a conic, of which S and H are the foci, is cut by two conjugate diameters in T, t; prove that the triangles SPT, HPt are similar to one another.

CHAPTER IX.

MISCELLANEOUS PROPOSITIONS.

ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN GEOMETRICAL CONDITIONS.

1. IF any five independent conditions be given, to which a conic is to be subject, each of these, expressed in algebraical language, will give an equation for the determination of the five arbitrary constants which the equation of the conic involves. Hence, five conditions suffice for the determination of the conic. It may, however, happen that some of the equations for the determination of the constants rise to a degree higher than the first, in such a case, the constants will have more than one value, and more than one conic may therefore be described, satisfying the required conditions, although the number will still be finite.

The geometrical conditions of most frequent occurrence are those of passing through given points and touching given straight lines, with such others as may be reduced to these. We proceed to consider how many conics may be described in each individual case.

2. Let five points be given.

In this case we have merely to substitute in the equation of the conic the co-ordinates of the several points for a, ß, y; we shall thus obtain five simple equations for the determination of the constants, and one conic only will satisfy the given conditions.

3. Let four points and one tangent be given.

Take three of the points as angular points of the triangle of reference. Let f, g, h be the co-ordinates of the fourth

given point, la + mẞ+ny = 0, the equation of the given tangent. Let the equation of the conic be

Then for the determination of the ratios λ: μ:v, we have the equations

λ μ V
+ +
g
h

0,

x2l2 + μ3m2 + v3n3 — 2μvmn — 2vλnl - 2xμlm = 0.

These equations will give two values for the ratios, and prove therefore that two conics can be described satisfying the required conditions.

4. Let three points and two tangents be given.

Take the three points as angular points of the triangle of reference. Let the two given tangents be represented by the equations

la + mB+ny = 0,
l'a+m2ß+n'y=0.

we have, for the determination of λ: :v, the equations

x2l2 + μ3m2 + v3n2 — 2μvmn -2vλnl - 2xμlm = 0,

λ2l'2 +μ3m”2 + v3n'2 — 2μvm'n' — 2vλn' l' — 2λμl' m' = 0, which, being both quadratics, give four values for each of the ratios, shewing that four conics may be described satisfying the given conditions.

5. Let two points and three tangents be given.

Take the three tangents as lines of reference, and let f, g, h; f', g', h', be the co-ordinates of the two given points.