47-49] CYCLIC REGIONS. 55 



circuits in one point only, and riot meeting .any of the n 2 

 remaining circuits. A barrier drawn in this manner does not 

 destroy the continuity of the region, for the interrupted circuit 

 remains as a path leading round from one side to the other. The 

 order of connection of the region is however diminished 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 

 n 1 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 an irreducible 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. This property is sometimes adopted as the definition of 

 an /i-ply-connected space. 



Irrotational Motion in Multiply-connected Spaces. 



49. The circulation is the same in any two reconcileable 

 circuits ABC A, A'B'C'A' drawn in a region occupied by fluid 

 moving irrotationally. For the two circuits may be connected by 

 a continuous surface lying wholly within the region ; and if we 

 apply the theorem of Art. 33 to this surface, we have, remembering 

 the rule as to the direction of integration round the boundary, 



/ (ABC A) + / (A'C'B'A') = 0, 



or 



If a circuit ABC A be reconcileable with two or more circuits 

 A'B'C'A', A"B"C"A", &c., combined, we can connect all these 

 circuits by a continuous surface which lies wholly within the 

 region, and of which they form the complete boundary. Hence 



/ (ABCA) + 1 (A'C'B'A') + / (A"C"R'A") + &c. = 0, 

 or / (ABC A ) = / (A'B'C'A') + 1 (A"B"C"A") + &c. ; 

 i.e. the circulation in any circuit is equal to the sum of the 



