The term ~iA In (s + c) in the expression for w as given in [42a] represents the potential 

 due to an image vortex of equal and opposite strength located at (-c,0), or at a distance A' 

 from the axis of the cylinder where h' = - c -x = yfc'^ +R'^~c. Thus hh' = R^, so that the 

 vortex and its image are located on inverse lines with respect to the cylinder. 



The stream function has the same sign as F or ^4 near the cylinder, the opposite sign 

 near the vortex. 



If the vortex is assumed to move with the fluid, it revolves around the cylinder at the 

 fixed distance h from its axis, in the opposite direction to that suggested by its own circula- 

 tion and with a linear velocity equal to tlie fluid velocity caused by the image vortex in the 

 cylinder, or with a velocity A/2c or r/inc. The formulas continue to represent the flow at 

 each instant provided the axes rotate with the vortex, as does also the image vortex in the 

 cylinder. 



The circulation around the cylinder can be changed by superposing the flow due to 

 another imaginary vortex located on the axis of the cylinder. Let the circulation due to this 

 vortex be F'. The total circulation around the cylinder is then F'- F, and thus vanishes if 

 F'= F. From [42b, c] and [40h,i], if r, Q are auxiliary polar coordinates with origin on the axis 

 of the cylinder and coordinate axis parallel to ar, as in Figure 58, potential and stream function 

 for the resultant flow are: 



F' 



c6 = -4(5, -(9,) -i- 0, </- = - 4 In-^ +— In r [42s,t] 



^ ^ in r 2n 



The added components of velocity are w'= - F'sin d/{2nr),v' = F ' cos d/{2nr). The added 

 term in w is (i/2n) F ' In (s-oTj), where s-x^ =re'^. 



If the original vortex is now assumed to move with the fluid, it revolves about the 

 cylinder as before but at a linear velocity F/(4»7'C) - F'/(47rA). The revolution is clockwise 

 if F'< A F/c, otherwise counterclockwise; if F'= A F/c, the vortex is stationary. 



C. A Cylinder of Radius R^ enclosing one of Radius R^ 



For this case, use is made of two circles lying on the same side of the origin of 

 coordinates. Let the circulation around the inner cylinder, in the space between the two, be 

 F. Let the a;-axis be drawn in the direction from the axis of the inner cylinder, called number 

 1, toward the axis of the outer, called number 2; and let their axes be at x^ and x^, respective- 

 ly. Let xjj have values ijj . and \p^ on the two cylinders, respectively. Then, using [42h] and 

 [421] for both cylinders, 



R. = - c csch — -, R„ = - c csch — =■ > ff, ,- [42t,u] 



* A A 



4c-^ + /? 2, x^ = yc2 + Rl >x^ f*2v,w] 



94 



