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adjoint algebraic analytic arbitrary circles circular numbers coefficients collinear commutative operators complex quantities conditions Si2 cone congruence conic conjugate consider constant converges coordinates corresponding cot ira cusp cyclic group defined denote derivatives determined discriminant locus ellipse equivalent expression factor finite norm formula fundamental Geometry given conditions Gramian Green's function group of order Hence homogeneous hyperplanes i-system identity infinite number integral curve intersection interval invariant inverse linearly independent log Q Mathematics method necessary and sufficient non-commutative operators non-twelve group number of nodes obtained octahedral group order 24 order of G osculating osculating circles osculating planes parameter pass pencil plane curve plane of distribution projective Projective Geometry prove quadratic transformation quotient group replaced respect satisfy the conditions satisfy the given singular singular solution SiS2 subgroup Substituting sufficient condition tangent tetrahedral group values vanish Wronskian zero