## Annals of Mathematics |

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2hxy 3-space Abel's theorem addition formula analytic angle apply approaches axis catenoid chain through MN circle coefficients commutative congruence conics conjugate with respect considered constant corresponding denoted derivatives differential equation double points E. H. Moore elliptic Elliptic integrals equal expression follows function geodesic given gives harmonic conjugate harmonic with MN helicoid Hence heptads hyperbolic inequalities infinite integral curves intersection invariant chain invariant points involution kF(k letter notation limit line coordinates linear linear fractional transformation mean velocity method obtain orthogonal pairs pencil plane point in common points harmonic points MN positive projective collineations projective geometry projectivities leaving proof quadric quintic real root represented satisfied second set sets of rails shadow bands shadow rails solution spaces Substituting surface tangent Theorem tion transformation tube values vanishes variable vertical whence write zero