included in it and exclude each other, and are called internal surfaces.
If the region has one bounding surface, we may suppose that surface to contract inwards without breaking its continuity or cutting itself. If the region is one of simple continuity, such as a sphere, this process may be continued till it is reduced to a point; but if the region is like a ring, the result will be a closed curve; and if the region has multiple connexions, the result will be a diagram of lines, and the cyclomatic number of the diagram will be that of the region. The space outside the region has the same cyclomatic number as the region itself. Hence, if the region is bounded by internal as well as external surfaces, its cyclomatic number is the sum of those due to all the surfaces.
When a region encloses within itself other regions, it is called a Periphractic region.
The number of internal bounding surfaces of a region is called its periphractic number. A closed surface is also periphractic, its number being unity.
The cyclomatic number of a closed surface is twice that of the region which it bounds. To find the cyclomatic number of a bounded surface, suppose all the boundaries to contract inwards, without breaking continuity, till they meet. The surface will then be reduced to a point in the case of an acyclic surface, or to a linear diagram in the case of cyclic surfaces. The cyclomatic number of the diagram is that of the surface.
19.] Theorem I. If throughout any acyclic region
,
the value of the line-integral from a point A to a point P taken along any path within the region will be the same.
We shall first shew that the line-integral taken round any closed path within the region is zero.
Suppose the equipotential surfaces drawn. They are all either closed surfaces or are bounded entirely by the surface of the region, so that a closed line within the region, if it cuts any of the surfaces at one part of its path, must cut the same surface in the opposite direction at some other part of its path, and the corresponding portions of the line-integral being equal and opposite, the total value is zero.
Hence if and are two paths from A to P, the line-integral for is the sum of that for and the closed path
