246 VORTEX MOTION. [CHAP. VII 



divide the velocity of A (say), viz. in.J(ir . AB), by the distance AO, 

 where 



and so obtain 



7T.AB* 



for the angular velocity required. 



If raj, ra 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 rotations be of opposite signs, lies in AB, or BA, 

 produced. 



If raj = ra 2 , is at infinity; in this case it is easily seen that 

 A, B move with constant velocity ra^Tr . AB) perpendicular to AB, 

 which remains fixed in direction. The motion at a distance from 

 the filaments is given at any instant by the formulae of Art. 64, 2. 



Such a combination of two equal and opposite rectilinear vortices 

 may be called a ' vortex-pair.' It is the two-dimensional analogue 

 of a circular vortex-ring (Art. 162), and exhibits many of the 

 characteristic properties of the latter. 



The motioD at all points of the plane bisecting AB at right 

 angles is in this latter case tangential to that plane. We may 

 therefore suppose the plane to form a fixed rigid boundary of the 

 fluid in 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 to the plane with the velocity m/27rd, where d is the 

 distance of the vortex from the wall. 



The stream-lines due to a vortex-pair, at distances from the vortices great 

 in comparison with the linear dimensions of the cross-sections, form a system 

 of coaxal circles, as shewn in the diagram 

 on p. 80. 



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 the figure, let EPD be 

 the section of the cylinder, A the position of 



the vortex (supposed in this case external), and let B be the 'image' of A 

 with respect to the circle EPD, viz. C being the centre, let 



