Variat ions on a Theorem of Abel

III. . . . . . . . . . . . . . . . . . . . . . . . . 321 Abel's Theorem in Original Form and Applications . . . . . . 327 (a) Origins of the Theorem and Abel's Proof . . . . . . . . . 327 (b) Inversion of the Trigonometric and Elliptic Integrals . . . . 335 (c) Singular Cubics-General Addition Theorems . . . . . . . 339 (d) The Poncelet Theorem . . . . . . . . . . . . . . . . . 345 Abel's Theorem and the Trace . . . . . . . . . . . . . . . 349 (a) Holomorphic and Meromorphic Forms on Analytic Varieties 349 (b) The Trace Mapping . . . . . . . . . . . . . . . . . . 351 (c) The G.A.G.A. Principle and Abel's Theorem in General Form 355 (d) Holomorphic Differentials and Rational Equivalence . . . . 359 (e) Abel's Theorem for Linear Spaces and Statement nfa Converse 364 Residues and Abel's Theorem . . . . . . . . . . . . . . . 368 (a) Local Properties of Residues; The Residue Theorem . . . . 368 (b) Residues on Projective Space and the Classical Abel Theorem 377 (c) Poincar4 Residues; the Proof of the Main Theorem . . . . . 380