54 IRROTATIONAL MOTION. [CHAP. Ill 



must be possible to connect them by a continuous surface, which 

 lies wholly within the region, and of which they form the complete 

 boundary ; and conversely. 



It is further convenient to distinguish between 'simple' and 

 'multiple' irreducible circuits. A 'multiple' circuit is one which 

 can by continuous variation be made to appear, in whole or in 

 part, as the repetition of another circuit a certain number of times. 

 A 'simple' circuit is one with which this is not possible. 



A 'barrier,' or 'diaphragm,' is a surface drawn across the 

 region, and limited by the line or lines in which it meets the 

 boundary. Hence a barrier is necessarily a connected surface, and 

 cannot consist of two or more detached portions. 



A ' simply-connected' region is one such that all paths joining 

 any two points of it are reconcileable, or such that all circuits 

 drawn within it are reducible. 



A 'doubly-connected' region is one such that two irreconcileable 

 paths, and no more, can be drawn between any two points A, B of 

 it; viz. any other path joining AB is reconcileable with one of 

 these, or with a combination of the two taken each a certain 

 number of times. In other words, the region is such that one 

 (simple) irreducible circuit can be drawn in it, whilst all other 

 circuits are either reconcileable with this (repeated, if necessary), 

 or are reducible. As an example of a doubly-connected region we 

 may take that enclosed by the surface of an anchor-ring, or that 

 external to such a ring and extending to infinity. 



Generally, a region such that n irreconcileable paths, and no 

 more, can be drawn between any two points of it, or such that n 1 

 (simple) irreducible and irreconciieable circuits, and no more, can 

 be drawn in it, is said to be ' n-ply-connected.' 



The shaded portion of the figure on p. 37 is a triply-con- 

 nected space of two dimensions. 



It may be shewn that the above definition of an /^-ply-connected 

 space is self-consistent. In such simple cases as n = 2, n = 3, this 

 is sufficiently evident without demonstration. 



48. Let us suppose, now, that we have an n-ply-connected 

 region, with n I simple independent irreducible circuits drawn 

 in it. It is possible to draw a barrier meeting any one of these 



