SMALL VIBRATIONS OF A HOLLOW VORTEX. 
187 
Therefore, substituting the value of x 0 already found 
rj.— — 
l( 6^0 1 1 j _ 2 
k~ l 
L—2 4 L-2 
' ( 2 ^o § 
j L + 5 
z»—i 
8 L — 2 ' l/_r4 L — 2 
^ \ £ 
L—2/ k 
for, since £ is at least of order W, the last term in the factor of £, can be neglected, 
and 
again 
Now 
Also 
since 
also 
therefore 
.L + 5, . k~ v . Ar 3 . 
a—— 8 L _2 ^+Tl-2 4 L-2^ 
^=A+K+ 2 ^(B 0 K*-B lK -*,]j-T|(B^-B 1 ^)6x 0 
1 d^ 
27r <7% 
= _A— 1 - 1_ 1 —— — 0 
4 L —2~ 4 L —2 
b^jaA+iAoL 
Ja, 1 Xt 0 
®1_1A A±li 1 
du iSR^ 2 °S 
t f=1(4 *-dw 
V-£ ( B lK -») =i(4a! _ 1 ) +i+i { i (4®-i)-i. i A 
=i(te 0 -i)-i f V=i 1 
4 L —2 2 L—2 
Hw- b a=-> 
—2&r 0 Aj^_2 & 
y =^-TB 0 F-TB#+AB s r* 
47T 
277 
P P 
P —1AA —_3. A — 
&2— 4 ^2^ 4 A 1 Ei 
2 B 2 
The principal term in B 3 is 
