138, 139.] RECTILINEAR VORTICES. 1G5 



found ; we have only to divide the velocity of A (say), viz. 



M 



j-p, by the distance AO, where 



7T . 



and so obtain ?? * | for the value required. 



7T . 



If m lt m 2 be of the same sign, i.e. if the directions of rotation 

 in the two filaments be the same, lies between A and B ; but 

 if the directions be of opposite signs, lies in AB, or BA, 

 produced. 



If m 1 = m z , is at infinity; in this case it is easily seen 



that A, B move with uniform velocity j-^ perpendicular to 



AB, which remains fixed in direction. The motion external to 

 the filaments at any instant is given by the formulae of Chapter IV, 

 Example 3. 



The motion at all points- of the plane bisecting AB at right 

 angles is tangential to that plane-. We may therefore suppose 

 this plane to form a fixed rigid boundary of the fluid on either 

 side of it; and so obtain the solution of the case where we 

 have a single rectilinear vortex in the neighbourhood of a fixed 

 plane wall to which it is parallel. The filament moves parallel 



tJ7 



to the plane with the velocity 05, where d is the distance of 

 the vortex from the wall. 



In the last case \rii^ = m^\ the stream-lines are all circles. 

 We can hence derive the solution of the case where we have a 

 single vortex-filament in a mass of fluid which is bounded, either 

 internally or externally, by a fixed circular cylinder. Thus, in 



Fig. 11. 

 p 



the figure, let DPE be the section of the cylinder, A the position 



