53 55.] MULTIPLY-CONNECTED SPACES. 49 



(simple) irreconcileable non-evanescible circuits, and no more, can 

 be drawn in it, is said to be ?i-ply-connected. 



The shaded portion of Fig. 3, Art. 40, is a triply-connected space 

 of two dimensions. 



It is shewn in note (B) that the above definition of an n-ply- 

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

 n = 3, this is sufficiently obvious without demonstration. 



54. Let us suppose, now, that we have an rz-ply-connected 

 region, with n 1 simple independent non-evanescible circuits 

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

 these circuits in one point only, and not meeting any of the n 2 

 remaining circuits*. A barrier drawn in this manner does not de 

 stroy the continuity of the region, for the interrupted circuit remains 

 as a path leading round from one side of the barrier to the other. 

 The order of connection of the region is however reduced by unity ; 

 for every circuit drawn in the modified region must be reconcileable 

 with one or more of the n 2 circuits not met by the barrier. 



A second barrier, drawn in the same manner, will reduce the 

 order of connection again by one, and so on ; so that by drawing 

 Ti1 barriers we can reduce the region to a simply-connected one. 



A simply-connected region is divided by a barrier into two 

 separate parts ; for otherwise it would be possible to pass from a 

 point on one side the barrier to an adjacent point on the other side 

 by a path lying wholly within the region, which path would in the 

 original region form a non-evanescible circuit. 



Hence in an n-ply-connected region it is possible to draw n 1 

 barriers, and no more, without destroying the continuity of the 

 region. We might, if we had so chosen, have taken this property 

 as the definition of an ?i-ply-connected space. We leave it as an 

 exercise for the student to prove that this definition is free from 

 ambiguity, and that it is equivalent to the former one. 



Irrotational Motion in Multiply- connected Spaces. 



55. The circulation is the same in any two reconcileable cir 

 cuits ABC A, A B C A drawn in a region occupied by fluid moving 



* In simple cases this is obvious. For a general proof see note (B). 

 L. 4 



