In oblique coordinates the general equation of a circle is x² + 2xy cos ω + y² + 2Ax + 2By + C = 0.
10. The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Circles are particular cases of conic sections; and 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.
| 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,
Oα·Oα′ = Oβ·Oβ′,