SMALL VIBRATIONS OF A HOLLOW VORTEX. 
165 
dzdp — dudv=-7 j - N 3 dudv 
r d(u.v) duV 
\dn) 
Therefore the expression to be made a minimum is 
whence 
In this put 
A/I I A/I H\_ n 
bu\p bu) bv\p bv 
r P=X\/p 
(i) 
and the equation in x becomes, remembering that 
^+^=0 
bu^bv* 
s X J /W , A°Y 
itt 2 tv 2 4, p 2 [\i?{/ / 
= 0 
The particular transformation employed for the Toroidal Functions makes 
M(^) + (^)l =sinh ' 8 “ =s "* 
whence 
o 
bu*^bv* 4S 2 * 
Put y=S~*R„ cos (nv+a), where R„ is a function of u only; then R ;i must satisfy 
d 2 R C dR , 2 nT) 
TV - a V-(« —i)R=0 
did S du ' 4 
which may be compared with the equation for Toroidal Functions, viz., 
rf 2 P,CrfP , 3 lvp _ 
«+s ! -D p =° 
It is easy to see that the equation in R is satisfied by 
