PROFESSOR W. M. HICKS ON VORTEX MOTION. 
37 
other component is everywhere perpendicular to these meridian planes. The vortex 
aggregates will be moving with rectilinear translation through the fluid with a 
velocity calculable, when the distribution of vortex motion is known, by Helmholtz’s 
method. Bring the aggregate to rest by impressing everywhere a velocity equal and 
opposite to the velocity of translation. The motion then consists of a flow up through 
the centre in the direction of previous translation, the fluid then streaming (in this 
most general case) in spirals round a certain circle. The circle may conveniently be 
called the equatorial axis of the aggregate. The line of symmetry through the 
centre in the direction of translation may then be termed the q')oIar axis. Whether 
we deal with ring-shaped or singly connected aggregates, the surfaces \(j will always 
be ring-shaped inside. In fact they are so also at the boundary, for the surface value 
of \fj really consists in the latter case of the outer boundary together with the 
polar axis. 
3 . Conceive now the aggregate divided up into a large number of ring-surfaces 
given by values of a paiameter xjj differing by dxjj, and confine attention to what is 
going on between the two surfaces \jj and xfj -f- dxjj. We shall suppose xp to increase as 
we pass from the outside inwards. Let dn denote the distance at a point between 
the surfaces \p and xjj + dip, dn to be measured also inwards. In the shell considered 
the lines of flow will be spiral, and the vortex-filaments also spirals, as indicated in 
the figure, the thin line P/’represeiiting a line of flow, the thick Pa a vortex-filament, 
Fig. I. 
and the line Pm a meridian section. Denote the velocity at P by v and the angle it 
makes with the meridian by (p. Also let w denote the molecular rotation at P, and y 
the angle the filament makes with the meridian—estimated positive when on the 
opposite side of the meridian to v. 
Consider the How between the two surfaces xp and xp -f- dxp across the “ parallel of 
latitude ” through P. The total flow must be the same for every parallel. The area 
through which the flow takes place is 2 TTpdn, where p is the distance of P from the 
