where 



T^ = [(x-c)^ + y^] , r2 = [(2; + c)2 + y2] 



and the angles 9^ and $2 are shown in Figure 53. The value of q is again given by Equation 

 [41h]. 



The equipotential curves are now the circular arcs ending at (- c,0), while the stream- 

 lines are the circles about these points; the equations of the arcs and circles, respectively, 

 are, from [41k, 1], 



a;2 +U + C cot— ■ j = c^ csc^ — = /?^2 j-^gf] 



X + c coth — j + y^ = c^ csch^^ = /?^2 |-^2g] 



where Ri, R, denote the corresponding circular radii. The points (- c,0) are inverse points 

 with respect to each circle; and the equation of any i// circle can also be written in terns of 

 geometrical quantities, from [4:1m], as 



In^ = ± sinh~i — [42h] 



li A > 0, iff has everywhere the opposite sign to x, whereas qS is many-valued. If is 

 assumed to be zero on U»e >r-axis wherever | ar | > c, it decreases in passing above the points 

 (- c,0), becomes - 77 .4 on the a^-axis between these points, and decreases further to - 2;7.4 on 

 returning below the points (- c,0) to the starting point, where \ x \ > c. Thus there is circula- 

 tion of magnitude V = 2 tt A about (c,0) and of magnitude -277- A about (- c,0). 



By inserting cylindrical boundaries along one or two of the t/f circles, a number of cases 

 of motion with circulation can be handled. In order to apply the formulas to a given case, it 

 is necessary to find .4 and c, and the location of the origin of coordinates, in terms of given 

 quantities. 



A. Two Circular Cylinders 



Two circular cylinders neither enclosing the other, with axes D distance apart, may be 

 represented by causing two of the i// circles to coincide with the circles representing the 

 cylinders. For example, the circles may be those labeled 1 and 2 in Figure 54, where number 1 

 encloses the point (c,0) and number 2 the point (-c, 0). Let the given circulation around any curve 

 encircling cylinder number 1 just once in the positive direction, but not cylinder number 2, be 

 r, and that around any curve encircling number 2 only, - F. Let i// have values i/j^ and il,^ on 

 the two cylinders, respectively. Draw the ^-axis from 2 toward 1, as in Figure 54. Then A =-- 



r/27T. 



In this case ip has the sign of A or F, and i/f^^ has the opposite sign; for, r^/r^ < 1 near 

 cylinder 2 and r /r > 1 near cylinder 1, If A > 0, <// increases from 1 to 2, corresponding to a 

 downward flow between the two cylinders. Hence, whatever the sign of A, 



92 



