71 
of Edinburgh, Session 1875 - 76 . 
tion varies [Y. M. § 60 (e)] from point to point of the length 
of the filament, and from time to time inversely as the area 
of the cross section. The product of the area of the cross section 
into the rotation is equal to the circulation or cyclic constant of 
the filament. 
24. Vorticity will be used to designate in a general way the 
distribution of molecular rotation in the matter of a vortex. Thus, 
if we imagine a vortex divided into a number of infinitely thin 
vortex filaments, the vorticity will be completely given when the 
volume of each filament and its circulation, or cyclic constant, are 
given; but the shapes and positions of the filaments must also be 
given in order that, not only the vorticity, but its distribution, can 
be regarded as given. 
25. The vortex density at any point of a vortex is the circula- 
tion 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. 
26. Divide a vortex into infinitesimal filaments inversely as their 
densities so that their circulations are equal; and let the circula- 
tion of each be — of unity. 
n 
Take the projection of all the fila- 
ments on one plane. — of the sum of the areas of these projections 
n 
is (Y. M. §§ 6, 62) equal to the component impulse 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. Bind, 
according to Poinsot’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). 
