twice-traversed segment GD is first separated into two parallel segments. Then the circula- 

 tion around this combined curve equals zero. But this circulation is the difference of the 

 circulations in the positive directions about DEF and GHK\ for GHK was traversed in the 

 negative direction, so that its contribution to /y ds was reversed in sign and the contribution 

 of GD, which is traversed twice but in opposite directions, cancels out. Hence the circulations 

 around DEF and GHK are equal. 



In the same way it can be shown that the circulation about a curve that encircles 

 several internal boundaries is the sum of the circulations around the separate boundaries. 



If the velocity potential at any point P near A is now defined by 



as in Section 6, its value for a path of integration such as PRPq in Figure 14 is easily seen 

 to exceed its value for a path such as PQPq by the circulation F around A. For, geometrically, 

 these two paths togetlier make up a closed curve encircling A. Other paths may encircle A in 

 the negative direction, or several times. Tims, if (f) is the value for one path, other paths of 

 integration may give any one of the values 



</) + nV 

 where n is any positive or negative integer. 



The potential is thus many valued in a multiply connected space; to each point P there 

 belongs an infinite number of values of 0, spaced F apart. The particle velocity as calculated 

 from is, however, single-valued, since all branches of the potential, characterized by various 

 values of n, have the same space derivatives. 



If several internal boundaries are present, the potential is many valued in a more com- 

 plicated fashion. In any case it follows from Equation [6g] that in going around any closed 

 curve in the positive direction the potential decreases by an amount equal to the circulation 

 around the curve. 



An alternative prodedure sometimes 

 adopted is to introduce enough imaginary 

 barriers extending to infinity so that, if these 

 barriers are never crossed by any path, the 

 integral defining cb remains single-valued. 

 Such a barrier is shown at ST in Figure 14. 

 But then discontinuities in d> may occur at the 

 imaginary barriers; and, if the velocity at a 

 point on a barrier is to be represented by 



derivatives of (f>, the barrier must be moved 



Figure 14 - Illustrating the definition of the 

 temporarily to one side. ■ . velocity potential in a doubly connected 



space. 



25 



