164 VORTEX MOTION. [CHAP. VI. 



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

 may be written 



2mM = 0, 2ratf = ..................... (36). 



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

 with regard to the time. Hence (36) express that the point whose 

 co-ordinates are 



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

 the centre of inertia of a mass 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 pro 

 jection may be called the axis of the system. 



139. We proceed to discuss some particular cases. 



(a) First, let us suppose that we have only one vortex-filament 

 present, and let have the same sign throughout its infinitely small 

 section. Its centre, as just defined, will lie either within the sub 

 stance of the filament, or at all events infinitely close to it. Since 

 this centre .remains. at rest, the filament as a whole will be station 

 ary, 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 



. 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 fila 

 ment is the same as in Art. 35. 



(6) Next suppose that we have two vortices, of strengths m , 

 m 2 , respectively. Let A, B be their centres, the centre of 

 the system. The motion of each filament is entirely due to 

 the other filament, and is therefore always perpendicular to 

 AB. Hence the two filaments remain always at the same 

 distance from one another, and rotate with uniform angular 

 velocity about 0, which is fixed. This angular velocity is easily 



