48 IRROTATIONAL MOTION. [CHAP. III. 



any one point of it to any other by an infinity of paths, each of 

 which lies wholly in the region. 



Any two such paths, or any two circuits, which can by continu 

 ous variation be made to coincide without ever passing out of the 

 region, are said to be mutually reconcileable. Any circuit which 

 can be contracted to a point without passing out of the region is 

 said to be evanescible/ Two reconcileable paths, combined, form 

 an evanescible circuit. If two paths or two circuits be reconcile 

 able, it 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 non-evanescible 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. 

 There is no distinction between simple and multiple evanescible 

 circuits. 



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 evanescible. 



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) non-evanescible circuit can be drawn in it, whilst all other 

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

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

 we may take that enclosed by 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 



