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



leave it as an exercise for the student to deduce the formulae of 

 that example from those of the present article. 



If we take the hyperbolas as the stream-lines, the portions of 

 the axis of x which lie beyond the points (+ c, 0) may be taken as 

 fixed boundaries. We obtain in this manner the case of a liquid 

 flowing from one side to the other of a rigid plane, through an 

 aperture of breadth 2c made in the plane ; but since the velocity 

 at the edges of the aperture is infinite, this kind of motion cannot 

 be realized with actual fluids. 



92. Example 8. Let 



z = A (w + O, 

 whence 





Along the stream-lines ^r = ATT, we have 

 x=A((j)-e*), y=Air. 



As &amp;lt;f&amp;gt; increases from oo through zero to + oo , x increases from 



x , reaches a certain maximum value, and then goes back to 



oo . The maximum value is readily found to be when (f&amp;gt; = 0, and 

 is A. Hence the portions of the straight lines y = + Air which 

 lie between x oo , and # = A, may be taken as fixed 

 boundaries. Let us next trace the course of a stream-line in 

 finitely near to one of the former ; say ^ = A (TT a), where a is 

 infinitesimal. This gives 



x = A ((j&amp;gt; + e* cos -v/r), y = A TT A y. -f- AOL e*, 



approximately. As &amp;lt;f&amp;gt; increases from oo , x increases, whilst y re 

 mains at first approximately constant and equal to A (TT a) ; when, 

 however x approaches its maximum value, y increases to the value 

 ATT.. As (j&amp;gt; increases beyond the value zero, x diminishes, whilst 

 the excess of y over ATT slowly but continuously increases. 



The formulae (51) express then the motion of a liquid flowing 

 from a canal bounded by two parallel planes into open space*. 

 We see, however, from (50), that the velocity at the edges of these 

 planes (where &amp;lt; = 0, ^ = TT) is infinite ; so that the motion 



* The above example is due to Helmlioltz, Phil. Mag. Nov. 1868. A drawing 

 of the curves = const., ^ = const., is given in Maxwell s Electricity and Magnetism, 

 Vol. i., Plate xii. 



