768 Dr. J. R. Wilton on the Solution of 



6 a. In the case o£ a problem somewhat similar to that of 

 motion in a viscous fluid, which it may sometimes be taken 

 to replace, namely, motion (not irrotational) in a perfect 

 fluid, with the condition that the velocit} r should vanish on 

 the boundary, the solution may be written down. 



Let V 2 f=2?=V 2 %, say. 



And let the cylinder bounded by the curve (1) be moving 

 with velocity components U and V, and rotating with 

 angular velocity co. Then, when = r, we have 



yjr = Vy - Yx - i&>0 2 + y 2 ) , 



B^/B # = — V — (ox, "dtyl'dy = U — coy. 



Thus, after reduction, we find 



+=x(*,y)-t {x(Xi, Yi) +**» y 2 ) } _ x - j T e (Y'|| -x'g$U 



Remembering that 

 we may write down the velocity in the form 



where %1 = % (X X , Y^. 



If it is possible to choose % so that this expression vanishes 

 at infinity, we obtain a solution for the case in which the 

 cylinder moves through the fluid at rest at infinity. (Cf. 

 §§5 and 6.) 



If in addition to making i/r vanish at infinity ^ satisfies 

 the equation V 4 % = 0, we have V 4 ^ = 0, and the analytical 

 form of the solution is precisely the same as in the case 

 of slow motion of viscous fluid with the same boundary 

 conditions. In the problem under consideration in the 

 present paragraph we are not limited to the case of slow 

 motions, but it must not therefore be supposed that the 

 solution may be applied to problems of viscous fluid motion 

 in which the velocities contemplated are not small. It is 

 only in exceptional cases that this is true. 



