71-72] RELATIVE STREAM-LINES. 95 



If we trace the course of the stream-line \//- = from $ = + oc to < = - oo , 

 we find that it consists in the first place of the hyperbolic arc q JTT, meeting 

 the lamina at right angles ; it then divides into two portions, following the 

 faces of the lamina, which finally re-unite and are continued as the hyperbolic 

 arc J? = f TT. The points where the hyperbolic arcs abut on the lamina are 

 points of zero velocity, and therefore of maximum pressure. It is plain that the 

 fluid pressures on the lamina are equivalent to a couple tending to set it broad- 

 side on to the stream ; and it is easily found that the moment of this couple, 

 per unit length, is |7rp<? 2 c 2 - Compare Art. 121. 



72. CASE II. The boundary of the fluid consists of a rigid 

 cylindrical surface rotating with angular velocity o> 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, then, with the same notation as before, 

 d-^r/ds will be equal to the normal component of the velocity of the 

 boundary, or 



d^lr dr 



j = a>r -j- , 



ds ds 



if r denote the radius vector from the origin. Integrating we 

 have, at all points of the boundary, 



i/r = Ja>r 2 + const (1). 



If we assume any possible form of o/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 : 

 1. If we assume 



<<lr = Ar*cos26=:A(x*-y*) (2), 



the equation (1) becomes 



(io> - A)x 2 + (Jo) + A) y 2 =C, 



which, for any given value of A, represents a system of similar 

 conies. That this system may include the ellipse 



we must have (|o> A) a 2 = (Jo> 4- A)b 2 , 



or A = io) . 



Hence f = i<0 . _ ((e . _ y.) (3), 



