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

510

ANALYTIC GEOMETRY

or, since the expression on the left is the development of the
determinant | ABCD I by the minors of the last row,
(4) ABCDI =0.
Conversely, if ABCD = 0 and not all four of the determinants A are zero, the planes (2) meet in a single point. For,
we can assume that A4 = 0. Then the first three planes meet
in a single point (3) and this point lies in the fourth plane,
since by hypothesis (4) holds.
Thus we have proved the theorem:
THEOREM 3. The four planes (2) meet in a single point if and
only if the determinant of their coefficients vanishes and not all
four minors A1, A2, A3, A4 are zero.+
If the normals to the planes (2) are all parallel to a
plane, the determinant IABCDI obviously vanishes, for then
Al = A2 = A A4 = 0 and the expansion of [ ABCD I by the
minors of the fourth column, namely:
- D1A1 + D2A2 - DA3 + DA4
has the value zero.
Conversely, if I ABCDI vanishes by virtue of the vanishing
of A1, A,, A,, A4, the normals of the planes (2) are, by Theorem
2, all parallel to a plane.
Consequently, we can combine Theorems 2 and 3 in the more
general, though less useful, theorem:
THEOREM 4. The four planes (2) meet in a single point or,
their normals are all parallel to a plane, if and only if the determinant of their coefficients vanishes.
Finally, we enumerate the cases which can occur when the
normals to the four planes are parallel to a plane. First, the
* Stated algebraically this theorem reads: The four equations (2)
are compatible and have, moreover, a unique solution, if and only if
ABOCDI = 0 and A1, A2, A3, A4 are not all zero. This theorem includes
the theorem of Ch. XVI, ~ 9, Ex. 8, and also its converse. It is to be
noted that it was geometric considerations which led us here to a proof
which covered the converse as well as the theorem.