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



Then at all points of this section 



-^ = velocity of the fluid in the direction of the normal, 

 = velocity of the boundary normal to itself, 



Integrating along the section, we have 



^= Vy + const .......... . .............. (39). 



If we take any possible form of -v/r, the equation (39) is the 

 equation of a system of curves each of which would by its motion 

 parallel to x produce the set of stream-lines defined by ty = const. 

 We give a few examples. 



(a) If we choose for ty the form Vy -f const., then (39) is 

 satisfied identically for all forms of the boundary. Hence the 

 fluid contained within a cylinder of any shape which has a motion 

 of translation only may move as a solid body. If, further, the 

 cylindrical space occupied by the fluid be simply-connected, this is 

 the only kind of motion possible. This is otherwise evident from 

 Art. 49 ; for the motion of the fluid and the solid as one mass 

 evidently satisfies the boundary conditions, and is therefore the 

 only solution which the problem admits of. 



A 



(b) Let ^ = sin 6 (Example 1) ; then (39) becomes 



A 



- sin 6 = Vr sin + const. 

 r 



In this system of curves is included a circle of radius a, provided 



A T7 



= Va. 

 a 



Hence the motion produced in an infinite mass of liquid by a 

 circular cylinder moving through it with velocity V perpendicular 

 to its length, is given by 



-& = - sin 6, 

 r 



which agrees with (34). 



(c) With the same notation as in Example 3 let us assume 



