82 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



already discussed in Art. 34, the motion is cyclic; the circulation 

 in any circuit surrounding the origin being 2?ryLt. 



82. Example 3. Let us write 



. , , x + iy a 

 d&amp;gt; + i&amp;lt;\lr = /jilog - f - . 

 & x + ly + a 



If r lt r 2 denote the distances of any point in the plane xy from the 

 points (+ a, 0), and O l} 2 the angles which these distances make 

 with the positive direction of x, we have 



x-\-iy a_ r^e i&1 



y + a r z e i * 

 and therefor 



The curves o/r = const., i. e. l 2 = const., are circles passing 

 through the points (+ a, 0) ; the curves &amp;lt;j&amp;gt; = const, are the system 

 of circles, orthogonal to these. Either of these systems of circles 

 may be taken as the equipotential curves, and the other system 

 will then compose the stream-lines. In either case the velocity 

 at the points (+ a, 0) will be infinite ; so that we must exclude 

 these points (e.g. by small closed curves drawn round them) from 

 the region to which the formulas apply, which thus becomes 



7^ 



triply-connected. If the curves = const, be taken as the stream- 



r* 



lines, the circulation in any circuit embracing the first only of 

 the above points is ZTT/JL ; that in one embracing the second point 

 only is QTT/JL ; whilst that in a circuit embracing both is zero. 



83. Example 4. Assume 



&amp;lt;/&amp;gt; + iyfr = fji log (x + iy a) (x + iy + a). 



If ? j, r 2 , 1} 2 have the same meanings as in the last example, 

 this gives 



$ = Vr l r 2 , ^ = /J ,(0 l + e 2 ] ....... . ....... (28). 



The curves ?\r 2 = const, are a system of lemniscates whose poles 

 are at the points (+ a, 0). The curves l + 2 = const., i. e. 



11 y 



arc tan - -- \- arc tan - = const., 

 x a x -f a 



