Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE PLANE

445

is a curved surface and the planes are plane surfaces. In generalizing the foregoing discussions we should, then, say:
An equation in x, y, z represents, usually,* a surface. The
surface consists of all those points and only those points whose
coordinates, when substituted for x, y, z in the equation,
satisfy it.
Shifting the point of view, we assume now that it is a surface, and not an equation, which is given. Then we should
say:
The equation of a given surface is an equation in x, y, z which
is satisfied by the coordinates of every point of the surface and by
the coordinates of no other point.
Problem 1. Find the equation of the sphere whose center
is at the origin and whose radius is a.
A point (x, y, z) lies on this sphere if and only if the square
of its distance from the origin is equal to a2:
X2 + y2 + z2 = a2.
Therefore, this is the required equation.
Problem 2. Find the equation of the plane which passes
through the axis of x and makes an angle of
30~ with the (x, y)-plane, as shown in Fig. 1. z
This plane intersects the (y, z)-plane in the o0 -
line whose equation in the (y, z)-plane is
z = tan 30~ y or z =- /V3 y.
FIG. I
But this equation, considered as an equation in
x, y, z, leaves x unrestricted; consequently, it represents in
space the given plane, i.e. it is the equation of the given
plane.
* An equation in x, y, z does not always represent a surface. For
example, the equation x2 + y2 =0 represents a line, namely, the z-axis;
the equation X2 + y2 + z2 = 0 represents just one point, the origin; and
the equation x2 + y2 + z2 + 1 = 0 represents no point whatsoever.