and since parallel right lines may be considered as meeting in a point at infinity.*

The familiar example of the circle will sufficiently illustrate to the beginner the nature of the diameters of curves of the second degree. He must observe, however, that diameters do not in general, as in the case of the circle, cut their ordinates at right angles. In the parabola, for instance, the direction of the diameter being invariable, while that of the ordinates may be any whatever, the angle between them may take any possible value.

142. The direction of the diameters of a parabola is the same as that of the line through the origin which meets the curve at an infinite distance.

For the lines through the origin which meet the curve at infinity are (Art. 136)

ax2+2hxy + by2 = 0,

or, writing for h its value √(ab),

{√(a) x+√ (b) y}" = 0.

But the diameters are parallel to ax + hy = 0 (by the last article), which, if we write for h the same value √(ab), will also reduce to √(a)x + √(b) y = 0.

Hence every diameter of the parabola meets the curve once at infinity, and, therefore, can only meet it in one finite point.

* Hence, a portion of any conic section being drawn on paper, we can find its centre and determine its species. For if we draw any two parallel chords, and join their middle points, we have one diameter. In like manner we can find another diameter. Then, if these two diameters be parallel, the curve is a parabola, but if not, the point of intersection is the centre. It will be on the concave side when the curve is an ellipse; and on the convex when it is a hyperbola.

143. If two diameters of a conic section be such, that one of them bisects all chords parallel to the other, then, conversely, the second will bisect all chords parallel to the first.

The equation of the diameter which bisects chords making an angle with the axis of x is (Art. 141)

(ax+hy+g) + (hx+by+ƒ) tan0 = 0.

But (Art. 21) the angle which this line makes with the axis is O' where

whence

tan O' ==

a+h tan
h+ b tan 0

b tan tan 0'+h (tan0+tan 0') + a=0.

H

And the symmetry of the equation shows that the chords making an angle ' are also bisected by a diameter making an angle 0. Diameters so related, that each bisects every chord parallel to the other, are called conjugate diameters.*

If in the general equation h=0, the axes will be parallel to a pair of conjugate diameters. For the diameter bisecting chords parallel to the axis of x will, in this case, become ax+g=0, and will, therefore, be parallel to the axis of y. In like manner, the diameter bisecting chords parallel to the axis of y will, in this case, be by +f=0, and will, therefore, be parallel to the axis of x.

144. If in the general equation c=0, the origin is on the curve (Art. 81); and accordingly one of the roots of the quadratic (a cos20+2h cose sine+b sin20) p2 + 2 (g cose +f sin 0) p=0 is always p=0. The second root will be also p=0, or the radius vector will meet the curve at the origin in two coincident points, if g cos 0+f sin 0-0. Multiplying this equation by p we have the equation of the tangent at the origin, viz. gx+fy=0.† The equation of the tangent at any other point on the curve, may be found by first transforming the equation to that point as origin, and when the equation of the tangent has been then found, transforming it back to the original axes.

p,

*It is evident that none but central curves can have conjugate diameters, since in the parabola the direction of all diameters is the same.

† The same argument proves that in an equation of any degree, when the absolute term vanishes the origin is on the curve, and that the terms of the first degree represent the tangent at the origin.

transform the equation to parallel axes through that point, and find the tangent at it. Ans. 9x5y0 referred to the new axes, or 9 (x − 1) = 5 (y − 1) referred to the old.

If this method is applied to the general equation, we get for the tangent at any point x'y', the same equation as that found by a different method (Art. 86), viz.

ax'x + h (x'y + y'x) + by'y + g (x + x') +ƒ (y + y) +c=0.

145. It was proved (Art. 89) that if it be required to draw a tangent to the curve from any point x'y' not supposed to be on the curve, the points of contact are the intersections with the curve of a right line whose equation is identical in form with that last written; and which is called the polar of x'y'. Consequently, since every right line meets the curve in two points, through any point x'y' there can be drawn two real, coincident, or imaginary tangents to the curve.*

It was also proved (Art. 89) that the polar of the origin is gx+fy+c=0. Now this line is evidently parallel to the chord gx+fy, which (Art. 139) is drawn through the origin so as to be bisected. But this last is plainly an ordinate of the diameter passing through the origin. Hence, the polar of any point is parallel to the ordinates of the diameter passing through that point. This includes as a particular case: The tangent at the extremity of any diameter is parallel to the ordinates of that diameter. Or again, in the case of central curves, since the ordinates of any diameter are parallel to the conjugate diameter, we infer that, the polar of any point on a diameter of a central curve is parallel to the conjugate diameter.

146. The principal properties of poles and polars have been proved by anticipation in former chapters. Thus it was proved (Art. 98) that if a point A lie on the polar of B, then B lies on the polar of A. This may be otherwise stated, If a point move along a fixed line [the polar of B] its polar passes through a fixed point [B]; or conversely, If a line [the polar of A] pass

* A curve is said to be of the nth class, when through any point n tangents can be drawn to the curve. A conic is therefore a curve of the second degree and of the second class but in higher curves the degree and class of a curve are commonly not the same.'

through a fixed point, then the locus of its pole A is a fixed right line. Or again, The intersection of any two lines is the pole of the line joining their poles; and conversely, The line joining any two points is the polar of the intersections of the polars of these points. For if we take any two points on the polar of A, the polars of these points intersect in A.

It was proved (Art. 100) that if two lines be drawn through any point, and the points joined where they meet the curve, the joining lines will intersect on the polar of that point. Let the two lines coincide, and we derive, as a particular case of this, If through a point O any line OR be drawn, the tangents at R' and R" meet on the polar of 0: a property which might also be inferred from the last paragraph. For since R'R", the polar of P, passes through 0, P must lie on the polar of O.

P

R"

And it was also proved (Ex. 3, p. 96), that if on any radius vector through the origin, OR be taken a harmonic mean between OR' and OR", the locus of R is the polar of the origin; and therefore that, any line drawn through a point is cut harmonically by the point, the curve, and the polar of the point; as was also proved otherwise (Art. 91).

Lastly, we infer that, if any line OR be drawn through a point O, and

R'

R

T'

P the pole of that line be joined to O, then the lines OP, OR will form a harmonic pencil with the tangents from 0. For since OR is the polar of P, PTRT" is cut harmonically, and therefore OP, OT, OR, OT" form a harmonic pencil.

Ex. 1. If a quadrilateral ABCD be inscribed in a conic section, any of the points E, F, O is the pole of the line joining the other

two.

Since EC, ED are two lines drawn through the point E, and CD, AB, one pair of lines joining the points where they meet the conic, these lines must intersect on the polar of E; so must also AD and CB; therefore, the line OF is the polar of E. In like manner it can be proved that EF is the polar of O, and EO the polar of F.

E

D

F

Ex. 2. To draw a tangent to a given conic A

B

section from a point outside, with the help of the ruler only.

Draw any two lines through the given point E, and complete the quadrilateral as

in the figure, then the line OF will meet the conic in two points, which, being joined to E, will give the two tangents required.

Ex. 3. If a quadrilateral be circumscribed about a conic section, any diagonal is the polar of the intersection of the other two.

We shall prove this Example, as we might have proved Ex. 1, by means of the harmonic properties of a quadrilateral. It was proved (Ex. 1, p. 57) that EA, EO, EB, EF are a harmonic pencil. Hence, since EA, EB are, by hypothesis, two tangents to a conic section, and EF a line through their point of intersection, by Art. 146, EO must pass through the pole of EF; for the same reason, FO must pass through the pole of EF: this pole must therefore be 0.

=

147. We have proved (Art. 92) that the equation of the pair of tangents to the curve from any point x'y' is (ax+2hx'y' +by'2+2gx'+2fy'+c) (ax2+2hxy+by2+2gx+2fy+c) = {ax'x+h (x'y + y'x) + by'y + g(x' + x) +ƒ (y' + y) + c}3. The equation of the pair of tangents through the origin may be derived from this by making x'=y'=0; or it may be got directly by the same process as that used Ex. 4, p. 78. If a radius vector through the origin touch the curve, the two values of p must be equal, which are given by the equation

2

(a cos2 + 2h cose sine + b sin2 ) p2 + 2 (g cose +ƒ sin0) p+c=0. Now this equation will have equal roots if e satisfy the equation

(a cos2+2h cos sin✪ + b sin20) c = (g cos 0 +ƒ.sin 0)2. Multiplying by p', we get the equation of the two tangents, viz. (ac − g3) x2 + 2 (ch — gf) xy + (bc − ƒ2) y2 = 0.

This equation again will have equal roots; that is to say, the two tangents will coincide if

or,

c (abc+2fgh — aƒ2 — bg2 — ch3) = 0.

This will be satisfied if c = 0, that is, if the origin be on the curve. Hence, any point on the curve may be considered as the intersection of two coincident tangents, just as any tangent may be considered as the line joining two consecutive points. The equation will have also equal roots, if

Now we obtained this equation (p. 72) as the condition, that the equation of the second degree should represent two right lines. To explain why we should here meet with this equation again,