Vol I - 20

Unilateral Surfaces.—It will be noticed that Moebius' sheet has one continuous edge. Also if we pursue any closed path, our direction of progress and a direction on the surface per pendicular to the former and pointing to the left may be taken as an indicatrix. Along some closed paths the latter will be reversed. A line number of boundaries (bilateralizing cross-sec tion), as we see by merely tracing a bilateraliz ing re-entrant section and then making a cross section that crosses the trace once between two points of one boundary.

Let our surface possess B bounding curves; let it become simply connected by virtue of b bilateralizing, s bound-severing, and j bound joining cross-sections. Thenj.---c-1—b—s, and B+s—(e-1—b—s)=1 or B-l-2s-Fb=c. Since b is not zero, the number of boundaries of a uni lateral surface will always be less than its con nectivity: B < c.

Two Types of Unilateral Surfaces.— Draw a line connecting two points on different bilateralizing re-entrant sections. Make the bound-joining cross-sections ab and oh imme diately to the right and left of it, and rejoin the rg-entrant sections along the small portions did 6b. The result will be a common re-entrant closely following such a path on its left will not close, as its beginning and end will be on oppo site sides of the path. An incision along the latter evidently leaves our surface connected. Thus, on a unilateral surface, at least one non dividing re-entrant section can be made. We shall call it a bilateralizing one. In fact, the number of bilateralizing re-entrant sections will be that of independent paths along which the indicatrix is reversed. This type of the re section. For follow by a line immediately to the left the re-entr4nt section from a to b, the cross-section from b to a, the other re-entrant section to a and, finally, the cross-section to b. It will be seen that this line is closed, and so is the corresponding edge of the whole incision.

After this process of uniting bilateralizing re-entrant sections has been repeated as often as possible, if b is even, no btlateralizing entrant sections remain; if b is odd, there will remain one. Accordingly, there are two types of unilateral surfaces: ( + = c, b even : B= 2 — c, when the surface is unbounded (B)), becomes 2, 0, —2,-4 . . .

—1 (2) B + 2 (s ) + 1 c, b odd: K = 2 c =1, —1, —3 ..., when B=0.

The above surface (Fig. 9) is of the first type, even if extended so as to lose its bound ary; K:). Steiner's surface, which is equiva

lent to the projective plane, is of the second type; K=1.

Steiner's the points of the projective plane from a centre C. On each pro jecting ray, whose length CP we call r, lay off the segment l+r from C. The line at infinity will thus furnish a circle of radius unity, whose diametrically opposite points represent the same point (at co ) of our plane. Now let the entire new surface, consisting of the ends of the seg ments laid off from C, be deformed into the plane area of this circle (Fig. 6). Cut the latter from C to E, roll it into a cone, putting CE on CE' (the edges of the incision on different deformation of the surface, may be made to show continuous curvature, and the apex of the cone may be made to coincide with the centre of the circle of juncture. This is Steiner's sur face (Fig. 11). By punching out its centre and cutting by a plane perpendicular to the double line we get three elementary surfaces. Hence K-1, also c=1, b =1, B = O.

Boy has devised similar surfaces and inves tigated the connection between the character istic and Gauss total curvature in such cases.

Two connected surfaces possessing the same number (B) of bounding-curves, the same con nectivity and, in case they are unilateral, the same number of bilateralizing re-entrant sec tions, can now be made simply connected by means of the same number of independent re entrant sections. After a correspondence has been decided upon between the bounding curves, sides). Join the edges of CE (creating a double line) and also, by adjacent points, the two basal circles of the cone. This second juncture, by we draw c —1 bound-joining cross-sections be tween pairs of corresponding ones and further establish a one-to-one correspondence between the points of the original bounding curves, also of these cross-sections. The resulting simply connected surfaces will have their boundaries corresponding point for point. Schoenflies has demonstrated that under these conditions be tween the points in the interior of the simply connected areas, too, a continuous one-to-one correspondence may be established. This proves the theorem of Jordan, to the effect that our surface is equivalent to the projective plane, which possesses any line as a bilateralizing section and is simply connected.