Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/105

paths, and no more, can be drawn to connect any two points, are said to be $n$-ply connected.

A few examples may be given. The region internal or external to the surface of a chain link or an anchor ring is a doubly connected region; a simple electric circuit, either internal or external to the conductor, is a doubly connected region; on breaking the circuit both regions become simply connected. A lake containing two islands is a triply connected region,^[1]. The region surrounding a gridiron is $n$-ply connected where $n$ is the number of the bars.

 

§ 64. Cyclic Motion.—The subject of connectivity derives its importance chiefly from its relation to the class of fluid motions known as cyclic. In a simply connected region, for all motions having a velocity potential, the latter, $\phi$, is a single-valued function, having at every point in the system a definite assignable value, varying continuously from point to point throughout the system. When the region is doubly connected this manifestly may not be the case, for if there is a circulation around an irreducible circuit it is evident that if we follow the variation of $\phi$ round such circuit we shall on arriving at the starting point have two conflicting values. Thus, referring to Fig. 35 when the radial lines are taken to represent $\phi =$ constant, we are unable to assign a progressive series of values to the $\phi$ lines that will be consistent.

Under these conditions $\phi$ is termed a cyclic function, and its value depends upon the datum point chosen for its zero and the number of times the path of integration has been taken round a circuit.

A physical conception of velocity potential under these circumstances is somewhat difficult, but if we revert to the dynamical hypothesis and regard the velocity potential system as the pressure system by which the motion is generated, we encounter at once the same difficulty in another form. Before we are able

1. An erratum published in volume 2 has been applied: "P. 85, line 8 from top, the cavity of the labyrinth of the ear is given as an example of a triply connected region; this is in error, delete." The original text reads: "The cavity of the labyrinth of the human ear is a triply connected region, as also is a lake containing two islands" (Wikisource contributor note)

85