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

474 ANALYTIC GEOMETRY
Problem 2. Find the equations of the curve of intersection
C of two circular cylinders of the same radius a, whose axes
are respectively the axes of x and y.
The equations of the two cylinders are
(8) y2 + 2 = 2, x2 + 2 = a2,
and these two equations, taken together, represent the curve C.
They are not, however, the simplest pair of equations possible. If x0, Yo, zo are any set of values of x, y, z satisfying them
simultaneously, i.e. if
yo2 + Z2 = a2, x2 + Z2 = a2
elimination of z0 gives the relation
Xo2- y2 = 0,
which says that x,, yo, zo also satisfy the equation
(9) x2 _ y2 = 0.
That is, the curve of intersection C of the cylinders (8) lies on
the surface (9).
Conversely, the surface (9) intersects each of the cylinders
(8) in the curve C and in no other points. For, if x0, yo, zo are
any values of x, y, z satisfying equation (9) and the first, say,
of equations (8), we have:
Xo2 -y02 = y02 + o2 = a2.
Elimination of yo gives the relation
x02 + Z = a2,
which says that x0, yo, z0 satisfy also the second equation of
(8), q.e.d.
We have proved, then, that the total intersection of any two
of the surfaces (8) and (9) is the curve C. Hence any two of
the equations (8) and (9) define C. A simpler pair than the
pair (8) is the combination of one of the equations (8) with the
equation (9), for example
(10) x2 + z a2 - _ y =0.