φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_
_l_}^* : φ_{_y_ _l_}^*.
_i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_})
when the subscript (_ik_) denotes the component of φ in the plane
contained by the lines other than (_ik_). Therefore the theorem is
proved.
We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ...
= 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...)
= 0
The general six-vector _f_ is composed from the vectors φ, φ^* in the
following way:—
_f_ = ρφ + ρ^* φ^*,
ρ and ρ^* denoting the contents of the pieces of mutually perpendicular
planes composing _f_. The “conjugate Vector” _f_^* (or it may be called
the complement of _f_) is obtained by interchanging ρ and ρ^*.