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absolutely continuous affinely connected affinely connected manifold algebraic algebraic curve alternative Amer arbitrary branch point coefficients components congruence consider contains continuous curve continuous function coordinates corresponding cubic defined definition denote derivatives determined differential equations differential invariants elementary function elements end-points equivalent exists expression finite number Fourier series function fundamental set given Hence identically independent inequality infinite integral interval invariants of order irreducible map kernel L. E. Dickson Lemma linear Math matrix multiply nearly everywhere necessary and sufficient normal tensors obtained paper pentagons permutation plane polygons positive solution proof properties prove quartic quintics reduces relations replaced respect satisfied scalar differential invariants sequence sequential limit point set of points sextic simple closed curve sufficient condition summable Suppose surface tensor theory transformation values vanish variables W. H. Young zero