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



ellipses, respectively; the distance between the common foci 

 being 2c. 



Now let us assume 



where G is some real constant. This makes &amp;lt;f&amp;gt; + ity a function of 

 x + iy ; and the value of ty is 



T/T = Ge ^cos %, 

 so that (39) becomes 



gTJ _ Q~ t l 



Ce-v cos f = Fc cos . ~ -- h const. 



In this system of curves is included the ellipse whose parameter rj 

 is determined by 



n TT e&amp;lt;n ~ e ~^ 



Ce-*= Fc. - . 

 If a, b be the semi-axes of this ellipse, we have, by (42), 



so that 



a b 

 Hence the formula 



V (a-b) 



(43) 



gives the motion of an infinite mass of liquid produced by an 

 elliptic cylinder whose semi-axes are a, b, moving parallel to its 

 major axis with velocity F. 



That the above formula makes the velocity zero at infinity 

 appears from the consideration that when TJ is large, dx and dy are 



of the same order as e^dn or &*, so that -~ , -f- are of the 



dx dy 



order 6~ 2l? , or 2 , ultimately, where r denotes the distance of any 

 point from the axis of the cylinder. 



If the motion of the cylinder were parallel to its minor axis, 

 the formula would be 



