By Felix Klein

This quantity introduces Riemann's method of multiple-value capabilities and the geometrical illustration of those features by means of Riemann surfaces. It concentrates at the form of capabilities that may be outlined upon those surfaces, limiting the therapy to rational features and their integrals demonstrating how Riemann's mathematical rules approximately Abelian integrals could be derived by means of reflecting upon the stream of electrical present on surfaces. This paintings is without doubt one of the top introductions to the origins of topological difficulties.

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**Example text**

On the two lower handles the directions are the same, and opposed to that on the upper handle. ] riemann’s theory. 47 points, two are at a and b, the third at c, and the fourth at the corresponding point on the under side. It is difficult to see the cross-points at a and b (Fig. 25) merely because foreshortening due to perspective takes place at the boundary of the figure, and hence both stream-lines which meet at the cross-point appear to touch the edge. If the streamings on the under side of the surface (along which the flow is in the opposite direction) are taken into account, any obscurity of the figure at this point will disappear.

Introductory remarks. 29 functions of position have neither infinities nor cross-points. I have therefore spoken provisionally of parts of surfaces only. But it is natural to enquire concerning the behaviour of these relations when the whole of any closed surface is taken into consideration. This is a question which is intimately connected with the line of argument presently to be developed; §§ 19–21 are specially devoted to it. § 7. Streamings on the Sphere resumed. Riemann’s general Problem.

In all these constructions it is clear, ab initio, that the vortex-points at z0 and z1 must have equal and opposite intensities. For similar reasons the total intensity of all the vortexpoints must always be zero, and thus the theorem that the sum of the logarithmic residues must vanish has been placed on a physically evident basis as regards the real, as well as the with the theory of “Doppelbelegungen” for which cf. Helmholtz, Pogg. Ann. (1853) t. lxxxix. pp. 224 et seq. (Ueber einige Gesetze der Vertheilung elektrischer Str¨ ome in k¨ orperlichen Leitern), and C.