SMALL VIBRATIONS OF A HOLLOW VORTEX. 
193 
Hence 
^+ u »(^) 0 f+/«=° 
Now (f) is the flow along any curve from a fixed point up to the point in question. 
Let us take the curve to be formed by a straight line from the centre in the plane of 
the ring (v=n) up to the surface ( u=u'), and then along the ring to the point (u, v). 
The first part will be a function of the time alone, and will therefore disappear with 
f(t) ; of the part along the ring, that due to the cyclic motion will be constant, and 
therefore the corresponding part in <£ will disappear. The part depending on the 
velocity of translation will be proportional to x, which will introduce a quantity 
proportional to x in <£. This will contain terms in cos v, which will not enter again. 
Hence x must be equated to zero, or the velocity of translation will not be affected. 
There remains only the part depending on the flow along the surface due to the 
motion x- This we proceed to find. Denoting it by </>, 
*,=r 
1 d% du dn r 
p du dn dv 
2« 2 M v / (2&) 
dv 
yiSR 
2cdil x /(21c)[ 
d R, ( 
du \/(C — c) 
sin nvdv 
?iS 
s qp 
R„ 
sin nvdv 
the principal part of which is 
domic 
CP 
+ K~i)TrH c °s nv-(-l) n ] 
The part of this, independent of cos nv, will disappear with f(t). 
Further, since U is multiplied by we must only take their lowest terms, which 
ClrC 
are independent of v. Finally then equating to zero the coefficient of cos nv 
Now 
therefore 
■M=0 
2-^ 0 da 
dk a? dk 
_U du 
u dk 
2 C 
MDCCCLXXXIV, 
