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

QUADRIC SURFACES

573

Three diameters in the relationship R2 are called conjugate
diameters, and three diametral planes in the relationship R3
are called conjugate diametral planes.
There are infinitely many sets of three diameters and three
diametral planes in the relationship R1. For, let E be an
arbitrary section of the ellipsoid by a plane through 0, and
let D1 and D2 be any two diameters conjugate in E. Then
the diametral planes Ml and M2 conjugate to D1 and D2 will,
by Th. 5, pass through D2 and D1 respectively. Finally, the
diametral plane, M3, conjugate to the diameter, D3, in which
M1 and M2 intersect will, by Th. 6, contain D1 and D2 and
hence must be the plane of D1 and D2.
The following theorems follow directly by application of
Theorems 2 and 1, respectively.
THEOREM 8. The diameters with the direction components
11, ml, n 1, 12, m n2, 13, m3, n3 are conjugate if and only if
112 + mzmg+ nln2
121 + m2r3 n2n3 0
131 + m3ml + n3nl =
a2 b2 c2
THEOREM 9. The diametral planes
Ax + Bly + Cz = O, Ax + B2y + C2 = 0, A3X + B3y + C 3z= 0
are conjugate if and only if
a2AA2 +- b'BlB2 + c2CC = 0,
a2A2A3 + b2B2B3 + c2C2C3 = 0,
a2A3A, + b2B3Bl + c2C3G = 0.
EXERCISES
State and prove for the following surfaces the theorems
analogous to Theorems 1, 2.
1. The unparted hyperboloid.