therefore the chord represented by the equation ga+fy=0 is bisected at the origin.

Hence, through any given point can in general be drawn one chord which will be bisected at that point.

140. There is one case, however, where more chords than one can be drawn, so as to be bisected, through a given point.

If, in the general equation, we had g=0, f=0, then the quantity g cose +f sine would be = 0, whatever were the value of ; and we see, as in the last article, that in this case every chord drawn through the origin would be bisected. The origin would then be called the centre of the curve. Now, we can in general, by transforming the equation to a new origin, cause the coefficients g and f to vanish. Thus equating to nothing the values given (Art. 134) for the new g and f, we find that the coordinates of the new origin must fulfil the conditions.

ax' +hy+g=0, hx' + by' +ƒ=0.

These two equations are sufficient to determine x and y', and being linear, can be satisfied by only one value of x and y; hence, conic sections have in general one and only one centre. Its coordinates are found, by solving the above equations, to be

In the ellipse and hyperbola ab – h' is always finite (Art. 137); but in the parabola ab- h = 0, and the coordinates of the centre become infinite. The ellipse and hyperbola are hence often classed together as central curves, while the parabola is called a non-central curve. Strictly speaking, however, every curve of the second degree has a centre, although in the case of the parabola this centre is situated at an infinite distance.

141. To find the locus of the middle points of chords, parallel to a given line, of a curve of the second degree.

We saw (Art. 139) that a chord through the origin is bisected if g cos 0+fsin 0 = 0. Now, transforming the origin to any point, it appears, in like manner, that a parallel chord will be

bisected at the new origin if the new g multiplied by cose + the new ƒ multiplied by sin00, or (Art. 134)

cos (ax' +hy+g) + sin 0 (hx' + by' +ƒ)=0.

This, therefore, is a relation which must be satisfied by the coordinates of the new origin, if it be the middle point of a chord making with the axis of x the angle 0. Hence the middle point of any parallel chord must lie on the right line.

cos (ax+hy+g) + sin 0 (hx + by +ƒ) = 0,

which is, therefore, the required locus.

Every right line bisecting a system of parallel chords is called a diameter, and the lines which it bisects are called its ordinates. The form of the equation shows (Art. 40) that every diameter must pass through the intersection of the two lines

M'

X

ax+hy+g=0, and hx+by+f=0;
but, these being the equations by
which we determined the coordinates
of the centre (Art. 140), we infer that
every diameter passes through the centre of the curve.
It appears by making

alternately0, and 90° in

=

-M

is the equation of the diameter

bisecting chords parallel to the axis of x, and that

hx+by+f=0

M

X

is the equation of the diameter bisecting chords parallel to the

axis of y.

*The equation (Art. 138) which is of the form by = − (hx +ƒ) ± R is most easily constructed by first laying down the line hx+by+f, and then taking on each ordinate MP of that line portions PQ, PQ', above and below P and equal to R. Thus also it appears that each ordinate is bisected by hx + by +f.

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,

U

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) tan 0 = 0.

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

whence

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

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 a will, in this case, become ax+9=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 cos*0+2h cos sin 0 + b sin*0) p2 + 2 (g cos 0 + ƒ sin0) 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 cose + 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.

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 then the terms of the first degree represent the tangent at the origin.

Ex. The point (1, 1) is on the curve

3x2 - 4xy + 2y2+7x - 5y - 3 = 0;

transform the equation to parallel axes through that point and find the tangent at it. Ans. 9x-5y=0 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.