81 8-i.] SPECIAL CASES OF MOTION. 83 



. 



or arc tan -= ^-^- r, = const., 



1 



are a system of rectangular hyperbolas, orthogonal to the above 

 system of lemniscates, and passing through their poles. A drawing 

 of both systems of curves is given by Lame*. 



The formulae (28) make the velocity infinite at the poles, 

 which must therefore be excluded from the region to which the 

 formulae apply. If the lemniscates be taken as the stream-lines, 

 the velocity-potential //, (O l + 6^ is a cyclic function; the circulation 

 in any circuit embracing one pole only is ZTT/JL, that in a circuit 

 embracing both poles is 



84. Example 5. Assume 



dw , 2 a , nnN 



-j- = ulog - .................... -. (29), 



dz z 4- a 



in the notation of Art. 82. Since 



dw d(j&amp;gt; . d^lr 



dz dx dx 

 this gives 



? 2 



If we suppose the region occupied by the fluid to have the axis 

 of x as a boundary, there will be no ambiguity in these values of 

 u, v. Moreover, u, v will be everywhere finite and continuous 

 except on the axis of x. When y = , and x&amp;gt; a, then 6^ = # 2 = ; 

 and when y = 0, x&amp;lt;-a, then t = B 2 = TT. In each case v = 0. 

 When y = 0, and a &amp;gt; x&amp;gt; - a, we have 6 l = TT, 2 = 0, and therefore 

 v = /ATT. We have thus the solution of the following problem : 

 An infinite mass of liquid bounded by an infinite rigid plane but 

 otherwise unlimited is initially at rest, and a strip of this plane of 

 breadth 2a is supposed detached from the remainder and suddenly 

 pushed inwards with velocity ^TT ; to find the motion produced in 

 the fluid. In the above formulae the rigid plane corresponds to 

 the axis of x, and the fluid lies to the negative side of the latter. 



It appears from (30) that at the edges of the strip u is infinite. 

 * Lerons sur les Coordonnees curvilignes, p. 223. 



62 



