Taking B at an infinitesimal distance from A all paths between them but those equivalent to the shortest one approach closed curves (Fig. 2). Hence there are c 1 independent closed curves on a surface of connectivity c. Conversely, an independent set of c 1 closed curves does not divide the surface (for this would give rise to an equivalence between those bounding any portion of it) and can readily be so connected with two points A, B, as to form (after slight changes) c-1 paths from A to B, in addition to which there is the direct line join ing these points.
Connectivity is often investigated by the method of sections. The latter are incisions of three types: (1) cross-sections between two points on the boundary. They may be bound severing, if drawn between points of the same bounding-curve, or bound-joining, if between different ones. The former increase, the latter diminish, the number of boundaries by unity; (2) re-entrant sections, along closed curves, each furnishing two new rims; (3) e- (sigma-) sections starting at a boundary point and end ing at a point of their own right or left edge. These contain a re-entrant section and a bound joining section, and increase the number of bounding-curves by unity. For exceptions to these statements see the paragraph on unilateral surfaces.
Limiting our investigations to surfaces any sufficiently small area of which may be con sidered simply connected, we may divide any one, or system of several, of them by a suffi cient number (q) of cross-sections into (say, e) elementary areas. Since cross-sections start at a boundary, we must give a boundary to closed surfaces by puncturing them, i.e., taking out an infinitesimal area somewhere. The difference e q then proves characteristic of our system of surfaces, and in fact is known as its char acteristic: K = e q.
To prove this, let a second division, by q' cross-sections, yield e' elementary areas. Super pose the tracings of both divisions and let there be t crossings of the proposed incisions. Then the e areas left whole by the first division will be cut q' t times by the second, or the e' areas of the second q t times by the first. Both sets of incisions thus furnish e q' t= e'+q-Ft parts, which proves the proposition. We also see that the characteristic of a system of surfaces is the sum of their individual char acteristics.
Any surface can be rendered simply con nected by means of 1 K cross-sections, for let the q cross-sections which divide it into e elementary areas be traced, and let them meet in v vertices. Consider this division as a map of e districts, the traces, counted from vertex to vertex, being its frontiers. Between any adjacent districts obliterate one frontier (there by also removing two vertices). Repeat this operation on the new map, etc., until but one district remains. By what we have proven, the totality of remaining frontiers then constitute 1K cross-sections.
On the other hand the c 1 nearly closed curves connecting A with B (see above) can readily be turned into cross-sections if we first draw a re-entrant section in a circle of diameter AB, thereby "puncturing* the surface. Hence,
on closed surfaces, c 1=1 K, or c =-- 2 K, and if we retain this formula, the connectivity of a system of m surfaces will prove to be the sum of the individual connectivities, diminished by the number (m 1) of junctures necessary to make one surface of the system: c= (m-1) =1 -I- (ci-1).
Kronecker's researches have led to an analytical expression for the characteristic of a closed analytical surface f(x, y, .7) = 0. Let f(x, y, .1) be negative in the interior of this surface, and consider the family of surfaces f(x, y, s) = '4. As X increases from x) to 0, the surface has no real part at first, then, through the stage of isolated points or curves, real surfaces will develop. An isolated point develops into an ellipsoidal surface, increasing K by 2, while a closed curve (without multiple points) becomes an anchor-ring, leaving K unchanged. This, or the opposite, may occur several times as the parameter increases. Also, double points of the surface may arise, in the neighborhood of which the surfaces re semble one- or two-sheet hyperboloids, chang ing from the one shape to the other as the double-point stage is passed. In each of these cases the increase of K is found to be fu In fn 2 sgn sgn (signum) meaning ± 1 fnfafa according as the determinant is positive or negative, and 0 if it is 0, and at at Of ()I ft fs 1 53, al at, etc., 112 bTay 'M 57.
being partial derivatives.
(1) The surfaces formed by rotation of the lemniscates 1(x a)" + a)" + A (Fig. 3) about the x-axis. For positive they present single sheets of ellipsoid connectivity (K = 2), for negative A, pairs of sheets of the same kind (K=4). Within an infinitesimal sphere about the origin the transition is from the one-sheet to the two-sheet hyperboloid, as A. decreases through zero. At a = a' the two sheets be come isolated points and vanish (K=0). (2) The surfaces formed by rotation of the same lemniscates about the y-axis. At a' they reduce to an isolated curve and vanish without changing the characteristic (K=0).
fu fitful Thus K becomes 2 Zsgn I fn fn fn I , the fah.s sum to be taken over all points of intersection of the three surfaces: fi=0, f.=0, fs= O. Moreover, this expression lends itself to trans formation into the integral by means of which Gauss represents the "total curvature) of the surface f(x, y, z)=0, so that we finally get: K total curvature. Connectivity of Riemann If w be an n-valued algebraic function of the com plex variable z (see COMPLEX VARIABLE), let all values of z be represented on a spherical sur face. Superpose radially n copies or sheets of this surface and imagine that for one value zo for which the n w-values are distinct, one value of w belongs to each so, i. e., to each of the n sheets. The values w, . . . w, will vary continuously with z, constituting n branches of the function w. For some values of z, how ever, say for . . . , but some among