38 
PROFESSOR W. HICKS OH VORTEX MOTIOX. 
polar axis. Hence 27Tpv cos (^cln is constant over the surface i//. It must therefore 
be of the form/(r//) drlf. So far i// is only defined as the parameter which detennines 
the particular surface. Choose the parameter so that /(i/;) = 1. if/ is then analogous 
to the stream-fimction in the simple case. It acts in fact as the stream-function for 
the component of velocity v cos cj). Similar reasoning leads to the conclusion that 
ojp cos yf/n is also of the form /(t//) dxp, say/Ic/i//. Hence 
2 Trpv cos (f)d}i — dxjj .(1) 
2 TTpoi cos ycZw — fdxjj .(2). 
We started with the supposition that the stream-lines and vortex-lines must lie on 
the same surfaces xjj. In other words, there must be no component rotation perpen¬ 
dicular to ifj. This may be ex])ressed in other words by the statement that the 
circulation round any circuit drawn wholly on ijj musi vanish. Take for this circuit 
any two parallels of latitude. Tire condition gives that the flow along one must 
equal the flow along the other. In other words, the flow round a parallel of latitude 
must be the same for all parallels on the same surface xjj. Hence 
27 rpv sin (^ = y.(3) 
where is a function of ifj. 
Equations 1, 2, 3 give conditions which any motion possible between any two given 
surfaces i// and rp -f- dxfj must satisfy. In our case, however, the motions in the 
.separate shells must fit together. We may regard the vortex-filaments as due to the 
velocities in t^vo successive shells, or as due to the different velocities on the inner 
and outer surfaces of the same shell—the velocities on the inner surface of one beino’ 
the same as on the outer of the next succeeding shell. If no\v be anv component 
of a filament, and dA the area perpendicular to Wi, the value of cuidA is given by half 
the circulation round dA. Apply this to the two components co cos y along a meridian 
and (o sin y along a parallel of latitude. As a circuit for w cos y take two parallels 
one on ip and the other on ip -j- dip. The flow along the first is 27 rpv sin ^ and along 
the latter 
2 Trpv sin (p -f- 277 y~ [pv sin (p) dn. 
Hence 
'jOI 
cos y . 27 Tpdn = — 277 — {vp sin <p) dn 
but by ( 3 ), 
Hence 
27 Tpv sin (p = f. 
Airpu) cos y = 
dn 
