88, 89.] MOTION WITHIN A ROTATING ELLIPTIC CYLINDER. 91 



Drawings of the curves (f&amp;gt; = const., ijr = const., in the above cases 

 are given in the Quarterly Journal of Mathematics, December, 1875. 

 These curves are the same for all confocal ellipses; so that the 

 formulaB hold even when the generating ellipse reduces to a 

 straight line joining the foci. In this case (44) becomes 



^r = Vce~v sin f (45), 



which would, except for the reasons stated in Art. 30, give the 

 motion produced in an infinite mass of liquid by an infinitely long 

 lamina of breadth 2c moving perpendicular to itself with velocity V. 

 Since however (45) makes the velocity at the edges of the lamina 

 infinite, this solution is destitute of practical value. 



When c = the problem of this section becomes that of (V) 

 above. The student may, as an exercise, work out the transform 

 ation of (43) into (34). 



89. Case II. The boundary of the fluid consists of a rigid 

 cylindrical surface rotating with angular velocity &&amp;gt; about an axis 

 parallel to its length. 



Taking the origin in the axis of rotation, and the axes of x, y 

 in a perpendicular plane, we have, with the same notation as 

 before, 



-j~ velocity in the direction of the normal to the boundary 

 ds 



dy dx 



= wy ~ cox -=- , 

 y ds ds 



(the component velocities of a point of the boundary being 



coy, cox, and the direction-cosines of the normal -~- , -j- J . 

 Integrating we have, at all points of the boundary, 



^ = - Jo, (x* + y*) + const (46). 



If we assume any possible form of -v/r, this will give us the 

 equation of a series of curves, each of which would by rotation 

 round the origin, produce the system of stream-lines determined 

 by ^. 



As examples we may take the following : 



