VOL. I - 20 the quantities w, .. . Wn will turn out equal. In these points we assume connection between the corresponding sheets, and denote them as branch-points. Such connection may not be feasible where other sheets intervene. In 4-space this difficulty would not arise. Limited as we are to 3-space, we may still suppose pas sage possible in these points between the sheets in question. Further, we find that whenever, starting at So, we take s in a loop (in all sheets simultaneously) about a branch-point, on re turning to zo the values to, . . . w will have undergone a permutation typical of that branch point. We prevent such loops, and render the branches single-valued, by means of incisions hrough all the sheets concerned, from z. to each branch-point. We further join every left edge of these incisions with the right one that exhibits the same w-values. This process (which, strictly speaking, again calls for a fourth dimension) completes our Riemann surface.

If we use a circular punch to cut out neigh borhoods of the m branch-points (through all the she Yts), the portion punched out at where first then /i0 . . sheets are con nected, will show n — (13n —1) — ( —1) . . . distinct simply connected parts. Thus all branch-points fwnish 1i'n— ;(/j;i— 1)] ele mentary areas. The neighborhood of zo, similarly punched out, yields n separate circles. The remainder falls into n elementary surfaces by means of m incisions from to the branch points, through all the sheets, therefore each counting for n cross-sections. Hence 2n -FiTn— 'Ai— 1)] — nm-=.2n — (Ay— 1). Granting that within a suffi ciently small neighborhood of every point P, any of the surfaces we consider has two sides (right and left) distinguished by the two per pendiculars to be drawn from P, it may happen that some continuous path on the surface start ing at P on the right side arrives at P on the left. side. The surface is then called unilateral; in the absence of such a possibility, bilateral. We have hitherto tacitly assumed the bilateral type for our surfaces.

Moebius called attention to the fact that a rectangular strip of paper aba'b', if its sides ba, a'b' be joined after a twist of as Fig. 4

directs, becomes unilateral. Moebius' sheet may conveniently be represented by folding the rectangular strip into triangular shape as in Fig. 5. The folds may be distinguished as posi tive or negative according as, on our way from ab to a'b', we pass from the lower to the upper sheet or the reverse. Each corresponds to a torsion or Positive folds will cancel against negative ones. Evidently a strip folded into the shape of a polygon of an even number of sides will thus represent a bilateral surface; if the number of sides be odd, a unilateral one. Ruled surfaces of the third order contain the Moebius sheet (Masckke). Closed surfaces without double points are bilateral.

entrant section, however, yields only one new bounding-curve. For only after completing a double circuit about the above closed path will the line following it on the left close in its turn, showing that the two edges of the incision blend into one. There also becomes possible a new kind of cross-section that leaves unchanged the Indicatrix.—The two half-normals at a point P, not being in the surface, are more conven iently replaced by a small circle about the point, taken in a definite (say counter-clockwise) ro tation about P. On the other side (of this point's neighborhood) the same rotation will be a clock wise one about P. Similarly, within the surface two infinitesimal perpendicular straight lines may be drawn, which if produced would form a right-handed Cartesian co-ordinate system (see ANALYTIC GEOMETRY) on one side (which we may define as the right one) and a left-handed one on the other. Such alternating contrivances are called indicatrices (Klein). They may be distinguished as right and left, or as positive and negative. If constructed continuously (i.e., without sudden transition to the opposite one) on continuous paths for all points that can thus be reached, one indicatrix will result for each point on a bilateral surface, while on a unilateral one a point will have both of them. Hence the term double surfaces for the latter type.