63-64] EXAMPLES. 79 



we see that the lines 0, 6 = ir/n are parts of the same stream- 

 line. Hence if we put n ir/a, we obtain the required solution in 

 the form 



- ft - Q 



<l> = Ar a cos , ^ = ^r a sin ............... (3). 



a ex. 



The component velocities along and perpendicular to r, are 



A I? ^~ 1 7r6 , . 7T i" 1 . 7T0 



A-r cos , and A -r sin ; 

 a a a a 



and are therefore zero, finite, or infinite at the origin, according as 

 a is less than, equal to, or greater than TT. 



64. We take next some cases of cyclic functions. 



1. The assumption 



w = p log z ........................... (1) 



gives = -/Alogr, ^ = -^0 ..................... (2). 



The velocity at a distance r from the origin is p\r\ this point 

 must therefore be isolated by drawing a small closed curve 

 round it. 



If we take the radii 6 const, as the stream-lines we get the 

 case of a (two-dimensional) source at the origin. (See Art. 60.) 



If the circles r const, be taken as stream-lines we get the 

 case of Art. 28 ; the motion is now cyclic, the circulation in any 

 circuit embracing the origin being 27174. TA_.$^> =. if 



2. Let us take 



z a 



/ox 

 (3). 



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

 from the points ( a, 0), and by 1} 2 , the angles which these 

 distances make with positive direction of the axis of x, we have 



z -f a = r 2 e z , 

 whence < = //, log rjr^ ty = yu, (0 1 



The curves </> = const., ^r const, form two orthogonal systems of 

 ' coaxal ' circles. 



