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



This system of curves includes the ellipse whose parameter is T 

 provided 



Ce-^ = \&amp;lt;oc\ 



or, using the values of a, b already given, 



C= 

 so that 



By the same reasoning as in the last article we see that at a great 

 distance from the origin the velocity is of the order - 3 . 



The above formulas therefore give the motion of an infinite 

 mass of liquid, otherwise at rest, produced by the rotation of an 

 elliptic cylinder about its axis with angular velocity &&amp;gt;. 



A drawing of the stream-lines in this case is given in the 

 Quarterly Journal of Mathematics, December, 1875. 



90. If w be a function of z, it follows at once from the defini 

 tion of Art. 74 that z is a function of w. The latter form of the 

 assumption is sometimes more convenient analytically than the 

 former. 



The relations (11) are then replaced by 



dx = dy dx _ dy 

 dcf&amp;gt; d^ ~cfy d$&quot; 



Also, since 



dw d(h . d-^r 



~T j + l j = u iv, 



dz dx dx 



we have 



dz 1 1 u . v 



j 



dw u iv q\q 



where q is the resultant velocity at (x, y). Hence if the properties 

 of the function ^ (= say,) be exhibited graphically in the manner 



already explained, the vector drawn from the origin to any point 

 in the plane of f will agree in direction with, and be in magnitude 

 the reciprocal of, the velocity at the corresponding point of the 

 plane of z. 



