In space of three dimensions there are two fundamentally different kinds of loci, of which the simplest examples are the plane and the straight line.

A particle which moves along a straight line has only *one degree of freedom*. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, by its distance from a fixed point on the line. If we take the line as our fundamental line \(\Lambda\) of Ch. I, the position of any of its points is determined by a single coordinate \(x\). A particle which moves in a plane, on the other hand, has *two* degrees of freedom; its position can only be fixed by the determination of two coordinates.

A locus represented by a single equation \[z = f(x, y)\] plainly belongs to the second of these two classes of loci, and is called a *surface*. It may or may not (in the obvious simple cases it will) satisfy our common-sense notion of what a surface should be.

The considerations of § 31 may evidently be generalised so as to give definitions of a function \(f(x, y, z)\) of *three* variables (or of functions of any number of variables). And as in § 32 we agreed to adopt \(f(x, y) = 0\) as the standard form of the equation of a plane curve, so now we shall agree to adopt \[f(x, y, z) = 0\] as the standard form of equation of a surface.

The locus represented by *two* equations of the form \(z = f(x, y)\) or \(f(x, y, z) = 0\) belongs to the first class of loci, and is called a *curve*. Thus a *straight line* may be represented by two equations of the type \(Ax + By + Cz + D = 0\). A *circle* in space may be regarded as the intersection of a sphere and a plane; it may therefore be represented by two equations of the forms \[(x – \alpha)^{2} + (y – \beta)^{2} + (z – \gamma)^{2} = \rho^{2},\quad Ax + By + Cz + D = 0.\]

Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance \(l\) apart (Fig. 18a). We can imagine the surface of the cylinder to be made up of a number of thin parallel rigid rods of length \(l\), such as \(PQ\), the ends of the rods being fastened to two circular rods of radius \(a\).

Now let us take a third circular rod of the same radius and place it round the surface of the cylinder at a distance \(h\) from one of the first two rods (see Fig. 18a, where \(Pq = h\)). Unfasten the end \(Q\) of the rod \(PQ\) and turn \(PQ\) about \(P\) until \(Q\) can be fastened to the third circular rod in the position \(Q’\). The angle \(qOQ’ = \alpha\) in the figure is evidently given by \[l^{2} – h^{2} = qQ’^{2} = \left (2a\sin\tfrac{1}{2} \alpha\right)^{2}.\] Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in Fig. 18b. It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings (Fig. 18c).

- We assume that the effects of the earth’s curvature may be neglected.↩︎

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