52 IRROTATIONAL MOTION. [CHAP. III. 



irreconcileable paths passing on opposite sides of the axis, e.g. 

 ACB, ADB in the figure. The portion of the plane zx for which 



Fig. 4. 



JL 



x is positive may be taken as a barrier, and the region is thus 

 made simply-connected. The circulation in any circuit meeting 



(&quot;2.TT 



this barrier once only, e.g. in A CBDA, is I - . rdO, or 2?j&amp;gt;t. That 



Jo r 



in any circuit not meeting the barrier is zero. In the modified 

 region &amp;lt; may be put equal to a single-valued function, viz. pO, 

 but its value on the positive side of the barrier is zero, that at an 

 adjacent point on the negative side is 2?ryLt. 



More complex illustrations of irrotational motion in multiply- 

 nected spaces will present themselves in the next chapter. 



58. Before proceeding further we may briefly indicate a some 

 what different method of presenting the above theory. 



Starting from the existence of a velocity-potential as the 

 characteristic of the class of motions which we wish to study, and 

 adopting the second definition of an n + 1-ply-connected region, 

 given in Art. 54, we remark that in a simply-connected region 

 every equipotential surface must either be a closed surface, or 

 else form a barrier dividing the region into two separate parts. 

 Hence, supposing the whole system of such surfaces drawn, we see 

 that if a closed curve cross any given equipotential surface once it 

 must cross it again, and in the opposite direction. Hence, cor 

 responding to any element of the curve, included between two 

 consecutive equipotential surfaces, we have a second element such 

 that the flow along it, being equal to the difference between the 

 corresponding values of $, is equal and opposite to that along the 

 former ; so that the circulation in the whole circuit is zero. 



If however the region be multiply-connected, an equipotential 

 surface may form a barrier without dividing it into two separate 

 parts. Let as many such surfaces be drawn as it is possible to draw 



