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1.5. Principally parametrized sphere congruences 19 2) Let f be an m-dimensional orthogonal net. Let f (1) , f (2) and f (3) be three of its Ribaucour transforms, and let three further orthogonal nets f (12) , f (23) and f (13) be given such that f (ij) is a simultaneous Ribaucour transform of f (i) and f (j) . Then generically there exists a unique orthogonal net f (123) that is a Ribaucour transform of f (12) , f (23) and f (13) . The net f (123) is uniquely defined by the condition that for every permutation (ijk) of (123) the corresponding points of f (i) , f (ij) , f (ik) and f (123) are concircular. The theory of discrete orthogonal nets will unify the theories of smooth orthogonal nets and of their transformations. 1.5. Principally parametrized sphere congruences The M¨ obius-geometric formalism is very convenient in description of hyper- sphere congruences. The classical case is, of course, that of two-parametric families of spheres in R3. Nonoriented spheres in R3 can be represented as elements of P(R 4,1 out ), where (1.60) R4,1 out = ˆ ∈ R4,1 : ˆ ˆ 0 is the space-like part of R4,1. Definition 1.21. (Principally parametrized sphere congruence) A map (1.61) S : R2 → nonoriented spheres in R3 is called a principally parametrized sphere congruence if the corresponding map ˆ : R2 → P(R4,1 out ) is a conjugate net, i.e., if for any lift of it to R4,1 out , (1.62) ∂1∂2ˆ ∈ span(ˆ s, ∂1ˆ ∂2ˆ). A principal parametrization exists and is unique for a generic congru- ence. The classical description of this is as follows. In an arbitrary paramet- rization of a congruence, consider two neighbors S(u + du1), S(u + du2) of a sphere S(u), obtained by infinitesimal shifts along both coordinate lines they intersect the original sphere along two circles C1(u) and C2(u). Thus, in the projective model of M¨ obius geometry of R3, based on the Minkowski space R4,1 of pentaspherical coordinates, these circles are de- scribed as L4,1 ∩ ( span(ˆ s, ∂1ˆ) )⊥ , resp. L4,1 ∩ ( span(ˆ s, ∂2ˆ) )⊥ i.e. their points are represented by elements ˆ ∈ L4,1 satisfying (1.63) C1 : ˆ ˆ = 0, ∂1ˆ ˆ = 0, resp. (1.64) C2 : ˆ ˆ = 0, ∂2ˆ ˆ = 0.