72 
Proceedings of the Royal Society 
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 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 pro- 
jections 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 negative portions. Thus 
their moments are not in general zero. 
Thus, according to these definitions, the resultant impulse of a 
vortex filament of infinitely small cross section and of unit 
circulation is equal to the resultant area of its curve. The 
resultant axis of a vortex is the same as the resultant axis of the 
curve, and the rotational moment is equal to the resultant areal 
moment of the curve. 
28. Consider for a moment a vortex filament in an infinite 
liquid with no disturbing influence of other vortices, or of solids, 
immersed in the liquid. We now see from the constancy of the 
impulse (proved generally in Y. M. § 19) that the resultant area, 
and the resultant areal moment of the curve formed by the 
filament, remain constant, however its curve may become con- 
torted ; and its resultant axis remains the same line in space. 
Hence, whatever motions and contortions the vortex filament may 
experience, if it has any motion of translation through space this 
motion must be on the average along the resultant axis. 
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