Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/104

§ 62 this and similar cases to suppose the source or sink to be circumscribed by a small closed curve, which in the case we have under consideration will be a circle. When we interchange the functions $$\psi$$ and $$\phi$$ the same considerations apply. In this case the space within the circular enclosure represents the section of a cylindrical filament, around which the cyclic motion of the fluid is taking place. The introduction of such an obstacle, i.e., a circumscribed area in a two-dimensional space or an infinite cylinder in a three-dimensional space, involves what is termed the connectivity of the region. Where no obstacle exists the region is said to be simply connected; where one or more such obstacles exist the region is multiply connected. The question involves certain points of definition.

§ 63. Connectivity.—It is possible to connect any two points in a region containing fluid by an infinite number of paths traversing the fluid. Such paths as can by continuous variation be made to coincide without passing out of the region are said to be mutually reconcilable.

Any circuit that can be contracted to a point without passing out of the region is said to be reducible.

Two reconcilable paths combined form a reducible circuit.

A simply connected region is one in which all paths joining any two points are reconcilable, or such that all circuits drawn within the region are reducible.

A doubly connected region is one in which two irreconcilable paths, and no more, can be drawn between any two points lying within it, so that any third path shall be reconcilable with the one or the other, or shall be in part reconcilable with one or the other, and in part reducible to the circuit formed by the two combined. (The latter portion of this definition is necessary to provide for the case of a third path being drawn making one or more circuits of the "obstacle.")

In general, multiply-connected regions, in which $$n$$ irreducible