Originally Titled: On Riemann's Theory Of Algebraic Functions And Their Integrals; A Supplement To The Usual Treatises - From The Unabridged Translation By Frances Hardcastle

Originally Titled: On Riemann's Theory Of Algebraic Functions And Their Integrals; A Supplement To The Usual Treatises - From The Unabridged Translation By Frances Hardcastle

CONTENTS:

PART I.
INTRODUCTORY REMARKS.
SECT. PAGE
1. Steady Streamings in the Plane as an Interpretation of the
Functions of x + iy 1
2. Consideration of the Infinities of w=f(z) .... 5
3. Kational Functions and their Integrals. Derivation of the
Infinities of higher Order from those of lower Order . 9
4. Experimental Production of these Streamings . . . 12
5. Transition to the Surface of a Sphere. Streamings on
arbitrary curved Surfaces . . . . . . 15
6. Connection between the foregoing Theory and the Functions
of a complex Argument 19
7. Streamings on the Sphere resumed. Riemann's general
Problem 21

PART II.
RIEMANN'S THEORY.
8. Classification of closed Surfaces according to the Value of
the Integer p 23
9. Preliminary Determination of steady Streamings on arbitrary
Surfaces 26
10. The most general steady Streaming. Proof of the Impossi-
bility of other Streamings 29
11. Illustration of the Streamings by means of Examples . . 32
12. On the Composition of the most general Function of Position
from single Summands 37
13. On the Multiformity of the Functions. Special Treatment
of multiform Functions 40
14. The ordinary Riemann's Surfaces over the x+iy Plane . 43
15. The Anchor-ring, p=I, and the two-sheeted Surface over
the Plane with four Branch-points 46
16. Functions of x+iy which correspond to the Streamings
already investigated 51
17. Scope and Significance of the previous Investigations . . 55
18. Extension of the Theory 56

PART III.
CONCLUSIONS.
19. On the Moduli of Algebraical Equations .... 59
20. Conformal Representation of closed Surfaces upon themselves 64
21. Special Treatment of symmetrical Surfaces .... 66
22. Conformal Representation of different closed Surfaces upon
each other 70
23. Surfaces with Boundaries and unifacial Surfaces ... 72
24. Conclusion 75