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1. The subject of investigation 2. The fundamental functions on a Riemann surface 3. The infinities of rational functions 4. Specification of a general form of Riemann's integrals 5. Certain forms of the fundamental equation of the Riemann surface 6. Geometrical investigations 7. Coordination of simple elements 8. Abel's theorem 9. Jacobi's inversion problem 10. Riemann's theta functions 11. The hyperelliptic case of Riemann's theta functions 12. A particular form of Riemann surface 13. Radical functions 14. Factorial functions 15. Relations concerning products of theta functions 16. A direct method of obtaining the equations relating theta functions 17. Theta relations with certain groups of characteristics 18. Transformation of periods 19. On systems of periods and on general Jacobian functions 20. Transformation of theta functions 21. Complex multiplication of theta functions 22. Degenerate Abelian integrals Appendix 1. On algebraic curves in space Appendix 2. On matrices.