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

SPHERES AND OTHER SURFACES 535
bola, the conjugate hyperbola, and the common asymptotic
lines; but this is true, also, of (5). We have, then, the following result.
THEOREM 2. The sections of two conjugate hyperbolic cylinders
by a plane M are two conjugate hyperbolas whose common asymptotes are the lines in which M cuts the common asymptotic planes
of the cylinders.
Returning to the general case, we assume that there is given
a second cylinder with vertical rulings, whose directrix, D, is
similar and similarly placed to D, or, if D is a hyperbola, is
similar and similarly placed to D or to the conjugate of D.
The equation of D, as a curve in K, can be written, according
to Ex. 40, p. 260, in the form:
(6) Ax2 + Bxy + Cy2 + Dx + Ey + F= 0.
The equation of the section S of the second cylinder by the
plane M is, then,
(7) Ax'2 + Bx'y' cos 6 + Cy'2 cos2 6 + Dx' + Ey' cos 0 + F = 0.
Since equations (5) and (7) fulfill the conditions of Ex. 40,
p. 260, it follows that S and are in the same relation as D
and D. We have thus proved, in the case in which D and D
are central conies, the following theorem.
THEOREM 3. If the directrices of two cylinders (with parallel
rulings) are similar and similarly placed conics, or if, in the case
of hyperbolas, each is similar and similarly placed either to the
other or to the conjugate of the other, then the sections of the
cylinders by the same plane or by two parallel plar es stand in like
relationship.
The converses of Theorems 1, 2, 3 are true, as is readily seen.
The three theorems and their converses can be stated equally
well in terms of projections. Thus Theorem 1 and its converse are equivalent to the theorem: A plane curve is a conic
of a certain type if and only if its projection on a plane not perpendicular to its plane is a conic of this type.