x2/α2 + y2/β2 + z2/O = 1,
(30)
with α2 + λ, β2 + λ, λ denoting the squares of the semiaxes of a confocal ellipsoid, and λ changed into μ and ν for a confocal hyperboloid of one sheet and of two sheets.
λ > 0 > μ > −β2 > ν > −α2,
(31)
then in the deformation of the hyperboloid, λ and ν remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so that
α2 + λ + β2 + μ + ν = OH2 = ½k2 (F − z) = ρ2 + OC2.
(32)
α2 + μ = ½k2 (z1 − z) = ρ2 − ρ12,
(33)