they have just been seen to have for their equations a particular class of equations of the second degree. Another particular class of such equations is that included in the form (Ax + By + C)(A′x + B′y + C′) = 0, which represents two straight lines, because the product on the left vanishes if, and only if, one of the two factors does, i.e. if, and only if, (x, y) lies on one or other of two straight lines. The condition that ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, which is often written (a, b, c, f, g, h)(x, y, I)2 = 0, takes this form is abc + 2fgh − af2 − bg2 − ch2 = 0. Note that the two lines may, in particular cases, be parallel or coincident.
Any equation like F1(x, y) F2(x, y) ... Fn(x, y) = 0, of which the left-hand side breaks up into factors, represents all the loci separately represented by F1(x, y) = 0, F2(x, y) = 0, ... Fn(x, y) = 0. In particular an equation of degree n which is free from x represents n straight lines parallel to the axis of x, and one of degree n which is homogeneous in x and y, i.e. one which upon division by xn, becomes an equation in the ratio y/x, represents n straight lines through the origin.
Curves represented by equations of the third degree are called cubic curves. The general equation of this degree will be written (*)(x, y, I)3 = 0.
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| Fig. 51. |
11. Descriptive Geometry.—A geometrical proposition is either descriptive or metrical: in the former case the statement of it is independent of the idea of magnitude (length, inclination, &c.), and in the latter it has reference to this idea. The method of coordinates seems to be by its inception essentially metrical. Yet in dealing by this method with descriptive propositions we are eminently free from metrical considerations, because of our power to use general equations, and to avoid all assumption that measurements implied are any particular measurements.
12. It is worth while to illustrate this by the instance of the well-known theorem of the radical centre of three circles. The theorem is that, given any three circles A, B, C (fig. 51), the common chords αα′, ββ′, γγ′ of the three pairs of circles meet in a point.
The geometrical proof is metrical throughout:—
Take O the point of intersection of αα′, ββ′, and joining this with γ′, suppose that γ′O does not pass through γ, but that it meets the circles A, B in two distinct points γ2, γ1 respectively. We have then the known metrical property of intersecting chords of a circle; viz. in circle C, where αα′, ββ′, are chords meeting at a point O,
where, as well as in what immediately follows, Oα, &c. denote, of course, lengths or distances.
Similarly in circle A,
and in circle B,
Consequently Oγ1·Oγ′ = Oγ2·Oγ′, that is, Oγ1 = Oγ2, or the points γ1 and γ2 coincide; that is, they each coincide with γ.
We contrast this with the analytical method:—
Here it only requires to be known that an equation Ax + By + C = 0 represents a line, and an equation x2 + y2 + Ax + By + C = 0 represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x2 + y2 + Ax + By + C, the equation of a circle is S = o. Let the equation of any other circle be S′, = x2 + y2 + A′x + B′y + C′ = 0; the equation S-S′ = 0 is a linear equation (S − S′ is in fact = (A − A′)x + (B − B′)y + C-C), and it thus represents a line; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points S = 0 and S′ = 0, therefore also S − S′ = 0); hence the equation S − S′ = 0 is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S″, = x2 + y2 + A″x + B″y + C″ = 0, the equations of the common chords are S − S′ = 0, S − S″ = 0, S′ − S″ = 0 (each of these a linear equation); at the intersection of the first and second of these lines S = S′ and S = S″, therefore also S′ = S″, or the equation of the third line is satisfied by the coordinates of the point in question; that is, the three chords intersect in a point O, the coordinates of which are determined by the equations S = S′ = S″.
It further appears that if the two circles S = 0, S′ = 0 do not intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S − S′ = 0; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the first-mentioned two circles.
13. The principle employed in showing that the equation of the common chord of two circles is S − S′ = 0 is one of very extensive application, and some more illustrations of it may be given.
Suppose S = 0, S′ = 0 are lines (that is, let S, S′ now denote linear functions Ax + By + C, A′x + B′y + C′), then S − kS′ = 0 (k an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be made to pass through any given point, say the point (x0, y0); if S0, S′0 are what S, S′ respectively become on writing for (x, y) the values (x0, y0), then the value of k is k = S0 ÷ S′0. The equation in fact is SS′0 − S0S′ = 0; and starting from this equation we at once verify it a posteriori; the equation is a linear equation satisfied by the values of (x, y) which make S = 0, S′ = 0; and satisfied also by the values (x0, y0); and it is thus the equation of the line in question.
If, as before, S = 0, S′ = 0 represent circles, then (k being arbitrary) S − kS′ = 0 is the equation of any circle passing through the two points of intersection of the two circles; and to make this pass through a given point (x0, y0) we have again k = S0 ÷ S′0. In the particular case k = 1, the circle becomes the common chord (more accurately it becomes the common chord together with the line infinity; see § 23 below).
If S denote the general quadric function,
then the equation S = 0 represents a conic; assuming this, then, if S′ = 0 represents another conic, the equation S − kS′ = 0 represents any conic through the four points of intersection of the two conics.
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| Fig. 52. |
14. The object still being to illustrate the mode of working with coordinates for descriptive purposes, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point O (fig. 52), we draw through O any two lines meeting the circle in the points A, A′ and B, B′ respectively, and then taking Q as the intersection of the lines AB′ and A′B, the theorem is that the locus of the point Q is a right line depending only upon O and the circle, but independent of the particular lines OAA′ and OBB′.
Taking O as the origin, and for the axes any two lines through O at right angles to each other, the equation of the circle will be
and if the equation of the line OAA′ is taken to be y = mx, then the points A, A′ are found as the intersections of the straight line with the circle; or to determine x we have
If (x1, y1) are the coordinates of A, and (x2, y2) of A′, then the roots of this equation are x1, x2, whence easily
| 1 | + | 1 | = −2 | A + Bm | . |
| x1 | x2 | C |
And similarly, if the equation of the line OBB′ is taken to be y = m′x1 and the coordinates of B, B′ to be (x3, y3) and (x4, y4) respectively, then
| 1 | + | 1 | = −2 | A + Bm′ | . |
| x3 | x4 | C′ |
We have then by § 8
|
x (y1 − y4) − y (x1 − x4) + x1y4 − x4y1 = 0, x (y2 − y3) − y (x2 − x3) + x2y3 − x3y2 = 0, |
as the equations of the lines AB′ and A′B respectively. Reducing by means of the relations y1 − mx1 = 0, y2 − mx2 = 0, y3 − m′x3 = 0, y4 − m′x4 = 0, the two equations become
| x (mx1 − m′x4) − y (x1 − x4) + (m′ − m) x1x4 = 0, x (mx2 − m′x3) − y (x2 − x3) + (m′ − m) x2x3 = 0, |
and if we divide the first of these equations by x1x4, and the second by x2x3 and then add, we obtain
| x { m ( | 1 | + | 1 | ) − m′ ( | 1 | + | 1 | ) } − y { | 1 | + | 1 | − ( | 1 | + | 1 | ) } + 2m′ − 2m = 0, |
| x3 | x4 | x1 | x2 | x3 | x4 | x1 | x2 |
or, what is the same thing,
| ( | 1 | + | 1 | ) (y − m′x) − ( | 1 | + | 1 | ) (y − mx) + 2m′ − 2m = 0, |
| x1 | x2 | x3 | x4 |
which by what precedes is the equation of a line through the point Q. Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing values, the equation becomes
that is,
