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, B, 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

X3l2 + μ3m3 + v3n2 — 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

lx+mẞ+ny = 0,

l'a+m'ẞ+n'y=0.

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

x2l2 + μ3m2 + v3n2 — 2μvmn

2

- 2vλnl - 2xμlm = 0,

X2l'2 + μ3m22 + 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.

Then, if the equation of the conic bet

+ β + γ - 2μνβγ - 2νλγα - 2λμαβ = 0,

we shall get, writing f, g, h; f', g', h', successively for a, ß, y, two quadratics for the determination of the ratios λ: μ:v, giving therefore four conics.

6. Let one point and four tangents be given.

Taking three of the tangents as lines of reference, the condition of touching the fourth given line gives a simple equation for the determination of the coefficients, and that of passing through the given point a quadratic. Hence, two conics may be described, satisfying the given conditions.

7. Let five tangents be given.

Taking three of the tangents as lines of reference, the condition of touching each of the others gives a simple equation for the determination of the constants, shewing that one conic only can be described satisfying these conditions.

The results of Arts. 5, 6, 7, may of course be deduced by the method of reciprocal polars, from those of Arts. 4, 3, 2.

8. Several other forms under which the data may be given, are reducible to a certain number of lines and points. Thus to have given a tangent and its point of contact is equivalent to having two points given, the points being indefinitely close together. Or, it may be regarded as equivalent to having two tangents given, these tangents being indefinitely nearly coincident. To have given that a conic is a parabola is equivalent to having a tangent given, since every parabola touches the line at infinity. To have given that it is a circle is equivalent to having two points given, since all circles intersect the line at infinity in the same two points. And this explains the reason why four circles can be described touching the sides of a given triangle, but only one circumscribed about it. So, to have given that a conic is similar and similarly situated to a given one is equivalent to having two points given. To have given an asymptote is equivalent to having two points given, for an asymptote may