45-47] CYCLIC REGIONS. 53 



infinite distance from S. Now the whole region occupied by the 

 fluid may be supposed made up of tubes of flow, each of which 

 must pass either from one point of the internal boundary to 

 another, or from that boundary to infinity. Hence the value of 

 the integral 



ffdy> 

 JJ dn 



taken over any surface, open or closed, finite or infinite, drawn 

 within the region, must be finite. Hence ultimately, when % is 

 taken infinitely large and infinitely distant all round from $, the 

 second term of (9) vanishes, and we have 



where the integration extends over the internal boundary only. 

 If the total flux across the internal boundary be zero, we have 



dn 

 so that (10) becomes 



simply. 



On Multiply-connected Regions. 



47. Before discussing the properties of irrotational motion in 

 multiply-connected regions we must examine more in detail the 

 nature and classification of such regions. In the following synopsis 

 of this branch of the geometry of position we recapitulate for the 

 sake of completeness one or two definitions already given. 



We consider any connected region of space, enclosed by bound- 

 aries. A region is 'connected' when it is possible to pass from 

 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 ' reducible.' Two reconcileable paths, combined, form a 

 reducible circuit. If two paths or two circuits be reconcileable, it 



