108 Sir William Thomson on Vortex Statics. 



25. The vortex-density at any point of a vortex is the 

 circulation of an infinitesimal filament through this point, 

 divided by the volume of the complete filament. The vortex- 

 density remains always unchanged for the same portion of 

 fluid. By definition it is the same all along any one vortex- 

 filament. 



2Q. Divide a vortex into infinitesimal filaments inversely as 

 their densities, so that their circulations are equal ; and let the 



circulation of each be - of unity. Take the projection of all 



the filaments on one plane. - of the sum of the areas of these 

 1 n 



projections is (V. M. § § 6, 62) equal to the component im- 

 pulse of the vortex perpendicular to that plane. Take the 

 projections of the filaments on three planes at right angles to 

 one another, and find the centre of gravity of the areas of 

 these three sets of projections. Find, according to Poin sot's 

 method, the resultant axis, force, and couple of the three forces 



equal respectively to - of the sums of the areas, and acting in 



lines through the three centres of gravity perpendicular to the 

 three planes. This will be the resultant axis ; the force 

 resultant of the impulse, and the couple resultant of the 

 vortex. 



The last of these (that is to say, the couple) is also called 

 the rotational moment of the vortex (V. M. § 6). 



27. Definition IV. — The moment of a plane area round any 

 axis is the product of the area multiplied into the distance 

 from that axis of the perpendicular to its plane through its 

 centre of gravity. 



Definition V. — The area of the projection of a closed curve 

 on the plane for which the area of projection is a maximum 

 will be called the area of projection of the curve, or simply the 

 area of the curve. The area of the projection on any plane 

 perpendicular to the plane of the resultant area is of course zero. 



Definition VI. — The resultant axis of a closed curve is a line 

 through the centre of gravity, and perpendicular to the plane 

 of its resultant area. The resultant areal moment of a closed 

 curve is the moment round the resultant axis of the areas of 

 its projections on two planes at right angles to one another, 

 and parallel to this axis. It is understood, of course, that the 

 areas of the projections on these two planes are not evanescent 

 generally, except for the case of a plane curve, and that their 

 zero-values are generally the sums of equal positive and nega- 

 tive portions. Thus their moments are not in general zero. 



