54 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
These numbers are graphically represented in fig. 1, Plate 1, where the abscissEe 
give E/Eo and the ordinates V/Vq. Dotted lines refer to points where calculation 
cannot be applied. On the same figure are placed outlines of the aggregates drawn 
to scale. Two things at once strike the eye. First, that the spherical aggregate 
evidently lies on the E .V curve of the rings, belongs, in fact, to the same family; 
and, secondly, that the variation of V with the energy is small over a very large 
range. The shape and nature of the aggregate when the energy is nearly that of the 
spherical form have not yet been determined. It is probable that as the energy 
diminishes the form lengthens along the polar axis, until when the euerg}^ is very 
small it becomes a long, thin, cylindrical aggregate. When this is so long that the 
end portions form only a small portion of the whole, it is possible to obtain an 
approximation to the energy, for when very long the fluid outside wiU be very 
nearly at rest (as in case of force outside a long helix). The velocity of propagation 
will then be the velocity at the axis. Let a be the radius of the cylinder, I its 
length. Then 
la~ = fc®. 
Again, if V denote the velocity along the axis, the velocity outside is zero, and the 
variation at the ends only a small part of the whole. Hence the circulation is 
given by 
/X = V/. 
Again let v denote the velocity at a distance r from the axis. Take a small 
rectangular circuit, h parallel to the axis, one inside distant r from the axis, the other 
outside. The circulation round this is bv. But it is also the value SwdA taken 
over the area of the rectangle. 
Therefore 
hv = Ji%2r clA = (volume) = .hn (a“ — r"), 
TT TT 
V = k{ar — ?■“), p, = Iha^ ; 
therefore 
Energy in E = 27rr. I dr . 
Jo 
TT/i,- p-/ .7 /-2\ 
TTfura- 
61 
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y 
IttcWI 
