which is the equation of the sphere, coordinates of the centre = (a, b, c), and radius = d.
A quadric equation wherein the terms of the second order are x² + y² + z², viz. an equation
x² + y² + z² + Ax + By + Cz + D = 0,
can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre −½A, −½B, −½C, and squared radius = ¼(A² + B² + C²) − D.
33. Cylinders, Cones, ruled Surfaces.—If the two equations of a straight line involve a parameter to which any value may be given, we have a singly infinite system of lines. They cover a surface, and the equation of the surface is obtained by eliminating the parameter between the two equations.
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x = mz, y = nz, the equations of a line of the system are x = mz + a, y = nz + b,—where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation ƒ(a, b) = 0: we have a = x − mz, b = y − nz, and the result of the elimination of the parameter therefore is ƒ(x − mz, y − nz) = 0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x = mz, y = nz. The equation of the section by the plane z = 0 is ƒ(x, y) = 0, and conversely if the cylinder be determined by means of its curve of intersection with the plane z = 0, then, taking the equation of this curve to be ƒ(x, y) = 0, the equation of the cylinder is ƒ(x − mz, y − nz) = 0. Thus, if the curve of intersection be the circle (x − α)² + (y − β)² = γ², we have (x − mz − α)² + (y − nz − β)² = γ² as the equation of an oblique cylinder on this base, and thus also (x − α)² + (y − β)² = γ² as the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the equations of a line are x − a = α(z − c), y − b = β(z − c), where α, β are functions of the variable parameter, or, what is the same thing, there exists between them an equation ƒ(α, β) = 0; the elimination of the parameter gives, therefore, ƒ[(x − a)/(x − c′), (y − b)/(z − c)] = 0; and this equation, or, what is the same thing, any homogeneous equation ƒ(x − a, y − b, z − c) = 0, or, taking f to be a rational and integral function of the order n, say (*)(x − a, y − b, z − c)n = 0, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*)(x, y, z)n = 0; and, in particular, (*)(x, y, z)² = 0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll; one on which they do is called a developable surface or torse.
Suppose, for instance, that the equations of a line (depending on the variable parameter θ) are x/a + y/c = θ (1 + y/b), x/a − z/c = (1/θ)(1 − y/b); then, eliminating θ we have x²/a² − z²/c² = 1 − y²/b², or say, x²/a² + y²/b² − z²/c² = 1, the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be remarked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter φ) are