where ^^ ^^ 



fj = [(a;-c)2 +y2] , r^ ^ [(x+ c)^ + y^ ] [41f,g] 



Here sin 6. = yA^, cos ^ = (a;-c)/rj, sin 6*2 = y/''2) <^os d^ = (x + c)/r^ from which tan (d„-dy) 

 may be found. Also 



g = (u^+v^f' =^£A. [41h] 



since 



2 



(a;2 -y^ -c^) +4 a;^ y2 = [ (a; + c)^ + y^j ^^i - c)^ + y'^] 



On the a;-axis, u = 2Ac/{x'^ -c^)\ on the y-axis, w = - 24c/(y2 +^2). 



Singularities occur at (c,0) and at (-c,0), where dw/dz -» «. 



The curves are circles with centers on the a;-axis, each enclosing one of the singular 

 points. The iji curves are circular arcs with centers on the y-axis and ending at the singular 

 points. The equations of these curves can be written either 



— = e^^^, 6,-6, =-4 [41iJ] 



( 4>\ 



Ix -c coth — I 



(y + c cot— j 



+ y^ = c^ csch2_ [41k] 



x^ + [y + c cot— ) = c^ csc^ -^ [411] 



The first of these equations in x and y is obtained'by substituting from [41f,g], squaring, 

 dividing by e^ , and rearranging. The second equation comes from the second expression 

 given for lA. 



The flow net is illustrated in Figure 54. 



The curve for </> = is the y-axis. If .4 > 0, -► + <» at (c,0) and - « at (-c,0). 



The function i{/ is many-valued with a period 27t \A\. As the point (x,y) goes positively 

 once around (c,0) without encircling (-c,0), 6, increases by 27t and t/j changes by AtA = - 2TrA; 

 if the point encircles (-c,0) instead, A 6^ = 2tt and AiA = 27tA. If, however, both singular 

 points are encircled, or neither, then Ai// = 0. 



The curve for i/> = (or - 27tA or 2irnA) is the ai-axis outside of (c,0) and (-c,0), provided 

 6. and 5, are measured from the same reference radius, as shown in Figure 53. Assume -4> 0. 

 Then successively smaller values of i/f are represented by the circular arcs above the axis 

 taken in descending order. On the a;-axis between (c.O) and (-c,0) 6. = rr, 6^ = 0, i/j - - nA. ~ 

 The arcs below the a;-axis represent successively smaller values of ip down to - 2TrA, for which 

 the curve is again the outlying part of the ar-axis. 



When is the potential, so that Equations [41d,e] hold, a uniform line source occurs 

 at each singular point, one being a positive source and the other an equal sink; if 4 > 0, the 



87 



