139, 140.] CIRCULAR VORTICES. 167 



AA at right angles may (either or both) be taken as fixed rigid 

 boundaries. We thus get the case where a pair of vortices, of 

 equal and opposite strengths, move towards (or from) a plane 

 wall, or where a single vortex moves in the angle between two 

 perpendicular walls. 



For other interesting cases of motion of rectilinear vortices 

 we refer to a paper by Professor Greenhill*. 



2. Circular Vortices, 



140. Next let us take the case where all the vortices present 

 in the fluid (supposed unlimited as before) are circular, having 

 the axis of x as a common axis. Let to- denote the distance of 

 any point P from this axis, S- the angle which TX makes with 

 the plane xy, v the velocity in the direction of -sr, and co the 

 angular velocity of the element at P. It is evident that u, v, co 

 are functions of x and &amp;lt;zr only, and that the axis of co is perpen 

 dicular to x, -cr. We have then 



y r C09--, z = TZ sn -, 



v = v cosS-, 10= usmS-A ,,,.... (37). 



= 0, 77 = - co sin ^, f = a&amp;gt; cos ^ j 



If we make these substitutions, writing TvdSrdxdi* for the volume- 

 element, in (30), and perform the integration with respect to ^, 

 we obtain 



T = 4-Trp //(CT u xv) ix co dx diz . 



The second and third of equations (31) are satisfied identically ; 

 the first gives 



/JW v co dx diff = . 



If we denote by m the strength codxd-5? of the vortex whose 

 co-ordinates are x, *&, these results may be written 



T 



3&amp;lt;m (TXU xv] or =-j (3$)&amp;gt; 



^im&v = - &quot;(39), 



* Quarterly Journal of Mathematics, 1877. 



