79 81.] SPECIAL CASES OF MOTION. 81 



velocity at the origin is infinite ; to avoid a physical absurdity we 

 must suppose the region to which our formulae apply to be limited 

 internally by a closed curve surrounding the origin. 



(d) If n = 2, each system of curves is composed of a double 

 system of lemniscates. The axes of the system &amp;lt;/&amp;gt; = const, coin 

 cide with x or y ; those of the system -fy const, bisect the angles 

 between these axes. 



(e) By properly choosing the value of n we get a case of 

 irrotational motion in which the boundary is composed of two rigid 

 walls inclined at any angle a. The equation of the stream-lines 

 being 



r n sin nO = const,, 



we see that the lines Q = 0, = - are parts of the same stream- 



n 



line. Hence we have only to put -- =a, and so obtain the required 

 solution in the form 



(j&amp;gt; = Ar- cos , -v/r = Ar a sin . 

 The component velocities along and perpendicular to r, are 



7T --1 7T# , A 7T --1 . 7T@ 



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. 



81. Example 2. The assumption 



W = fL log Z, 



or &amp;lt; + i-jr = /JL log re i& , 



gives &amp;lt;f&amp;gt; = /JL log r, ty = pQ. 



The equipotential lines are concentric circles about the origin ; 

 the stream-lines are straight lines radiating from the origin. Or, 

 we may take the circles r = const, as the stream-lines, and the 

 radii 6 = const, as the equipotential lines. In both cases the 



velocity at a distance r from the origin is - ; we must therefore 



suppose the origin excluded (e.g. by drawing a small circle round 



it) from the region occupied by the fluid. In the second case, 



i, 6 



