154-155] RECTILINEAR VORTICES. 245 



and, by the same reasoning, 



ffv{da;dy = ........................... (9). 



If as before we denote the strength of a vortex by m, these results 



may be written 



2wM = 0, 2rav=0 ........................ (10). 



We have seen above that the strength of each vortex is constant 

 with regard to the time. Hence (10) express that the point whose 

 coordinates are 



rp - _ /?/ - _ \ 



VO ^ j (j ^&amp;gt; j 



2m Zra 



is fixed throughout the motion. This point, which coincides with 

 the centre of inertia of a film of matter distributed over the plane 

 xy with the surface-density f, may be called the centre of the 

 system of vortices, and the straight line parallel to z of which it 

 is the projection may be called the axis of the system. 



155. Some interesting examples are furnished by the case of 

 one or more isolated vortices of infinitely small section. Thus : 



1. Let us suppose that we have only one vortex-filament 

 present, and that the rotation f has the same sign throughout its 

 infinitely small section. Its centre, as just defined, will lie either 

 within the substance of the filament, or at all events infinitely 

 close to it. Since this centre remains at rest, the filament as a 

 whole will be stationary, though its parts may experience relative 

 motions, and its centre will not necessarily lie always in the same 

 element of fluid. Any particle at a finite distance r from the 

 centre of the filament will describe a circle about the latter as 

 axis, with constant velocity m/7rr. The region external to the 

 filament is doubly-connected ; and the circulation in any (simple) 

 circuit embracing the filament is 2m. The irrotational motion of 

 the fluid external to the filament is the same as in Art. 28 (2). 



2. Next suppose that we have two vortices, of strengths m lt 

 m 3 , respectively. Let A, B be their centres, the centre of the 

 system. The motion of each filament as a whole is entirely due 

 to the other, and is therefore always perpendicular to AB. Hence 

 the two filaments remain always at the same distance from one 

 another, and rotate with constant angular velocity about 0, which 

 is fixed. This angular velocity is easily found; we have only to 



