0^ and i//^ are the potential and stream function for still another possible type of flow; for, as 

 was shown for d> in Section 7, (b^ and xh^ will satisfy Equations [13h], [13i] and [13f, g]. 

 Instead of x, y ov z may be substituted in both derivatives. 



It should be reniarl-;ed that when the older convention mentioned in Section 6 is employed 

 for the sign of the potential, the stream function t// is measured by the volume of fluid crossing 

 a curve in the clockwise direction, so that in a given case differences in the values of ih have 

 opposite signs, and the signs before the derivatives in Equations [13a, b] and [13j, ii] are 

 reversed. The positive direction for q^ in Equation [13c] is likewise reversed. The relation 

 between </> and i// as stated in [13f, g], however, remains the same. The simplest way to 

 summarize the difference between the two conventions is to say that all velocities are reversed 

 when a change is made from one to the other. 



14. TWO-DIMENSIONAL FLOW IN MULTIPLY CONNECTED SPACES 



In cases of two-dimensional flow, boundaries often occur which have finite dimensions 

 in directions parallel to the planes of motion. These are called internal boundaries. They 

 have the physical form of cylinders of unlimited length and are represented on the ccj'-plane 

 by closed curves, which may or may not be circular. The presence of an internal boundary 

 makes the space doubly connected; more generally, if two or more separate inner boundaries 

 occur, it is multiply connected. 



In irrotational motion, the circulation is required to vanish only around closed curves 

 which do not surround any inner boundary and hence can be contracted continuously down to a 

 point, in accordance with tlie explanation in Section 5. An example is curve C in figure 13, 

 where A represents an obstacle with a boundary that the fluid cannot penetrate. Around a 

 curve that encircles .4, such as DEF in Figure 13, the circulation may or may not vanish. 



Let tlie positive direction along all curves that encircle boundaries be chosen in the 

 same direction; it will be convenient to adopt the convention that, as a point traverses such 

 a curve positively, its projection on the aiy-plane eventually goes round the boundary 

 in the counterclockwise direction, or in the 

 direction of rotation from the .r-axis toward 

 the y-axis. Then the circulation has the same 

 value for all closed curves that encircle A 

 just once and do not encircle any other finite 

 'boundary. 



To show this, let DEF and GHK be two 

 such curves, and introduce a connection GD 

 between them, as illustrated in Figure 13. 

 Then the combined curve DEFDGKHGD, 



traced in this order, can be collapsed con- ^j^^^^ ^3 _ ^^^^^^ j^ ^ ^^^^^^ connected 



tinuously to a point; to do this, the space. 



24 



