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

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ANALYTIC GEOMETRY

The centers of the sections lie, then, on a line through the
center of the ellipsoid. Such a line is known as a diameter.
Accordingly, we can state our result as follows.
THEOREM 1. The locus of the centers of parallel sections of
the ellipsoid (1) is that portion of a diameter which lies within
the ellipsoid. If the direction components of the normals to the
planes of the sections are A, B, C, those of the diameter are a2A,
b2B, c2C.
Exercise. The tangent planes at the extremities of a
diameter are parallel to the sections whose centers the diameter contains.
Problem 2. Find the locus of the mid-points of a set of
parallel chords of the ellipsoid (1).
Let X,,u, v be the given direction cosines of one of the chords
and let P: (X, Y, Z) be the mid-point of the chord. Then
equations (2) represent the chord parametrically.
Since P is the mid-point of the chord, the algebraic distances, rl and r2, from it to the end points of the chord are
negatives of each other: r1 + r2 = 0. Hence, as in Problem 1,
we obtain the equation (4). But, whereas in that problem
X, /, v were auxiliary variables, here they are given constants,
and hence (4) is the equation satisfied by the point P of the
locus. But (4) represents a plane through the center of the
ellipsoid. Such a plane is known as a diametral plane. Thus
our result is:
THEOREM 2. The locus of the mid-points of a set of
parallel chords of the ellipsoid (1) is that portion of a diametral plane lying within the ellipsoid. If the direction components of the chords are 1, m, n, the equation of the diametral
plane is
lx + By +nz =
2 b2 C2