195] VORTEX PAIRS. 519 



fluid. Since either vortex moves with a velocity - - and half way 



between them the velocity being due to both is - = ^ we find 



~~ 



that a single vortex near a plane wall moves parallel to it with a 

 velocity one fourth that of the water at the wall. This is an 

 illustration of the method of 

 images, of frequent application 

 in hydrodynamics. 



As another illustration con- 

 sider the motion of a single 

 vortex-filament in a square corner 

 inclosed by two infinite walls. 

 The motion is evidently the same 

 as if we had a pair of vortex- 

 couples formed by the given Q A O 

 vortex and its images in the 

 two walls, turning as shown in 



Fig. 163 and forming what may p . 16g 



be called a vortex kaleidoscope. 



From the symmetry it is evident that the planes of the walls are 

 stream -planes, so that we may consider the motion in one corner 

 alone. If x and y be the coordinates of the vortex considered, we 

 have as due to the others, 



Y. K y Y. x* 



Zn x*-\-y* 2 it y (x 2 -f 



102) 



xx x Y, y* 



' ' *~* = ~ * 



Since u and v are the velocities -jr>. -~ of the vortex, we have for 



at at 



the equation of its path 



dx 



irvoN dt u x s dx 



103) -j = = -- - = , or 



ay v y ay 



dt 



dx _ dy 

 !c*~ ~~y*' 

 whose integral is 



111 T 12 ?/ 2 



104 ) + = ' ** + f = -' 



= -, sin 2 & cos 2 ^, 



n" ' 



a 



and in polar coordinates, 



105) 

 the equation of a Cotes's spiral, having one of the axes as an asymptote. 



