SMALL VIBRATIONS OF A HOLLOW VORTEX. 173 
Let \jj Q be the constant value over the surface of the tore. Then, dashed letters 
denoting the values of the functions on the tore, 
\ AJRL=i// 0 ( cos 6) cos nddO 
= V * < b 7 Eq. 3) 
but 
Hence 
This is more convergent than 2JT„, and is therefore convergent. 
Let [i denote the cyclic constant, then by (4) 
7rA 0 B' 0 = 2 \jj 0 \/ 2T' 0 
^= JYfl 3 J T 'olY 
7T\/ (C —c) [ °R 
T 'a R w 
• • (13) 
, 22 "_A_.L1 
^ R' 0 + lR'J • 1 * 
• • • (14) 
When the section of the tore is small compared with the aperture, the value of /x, 
correct to the fourth power of k, is 
/*=- 
27rvjr. 
^+( 1 -i(T^> 3 +< 2L - 1 +A(T^H- • • < 15 > 
6. Stream f unction for translat ion .—In the preceding case the conditions were that 
\fj must be constant over the tore and finite at an infinite distance from it. In the 
present case i// must be finite at an infinite distance and = ^Y p 2 over the surface, Y 
being the velocity of translation, and r ft the stream function for the tore moving in the 
fluid, at rest at infinity and referred to its instantaneous position. But if this condi¬ 
tion be applied, we shall also, on account of the cyclosis, obtain besides an added cyclic 
motion through the aperture determined by the surface condition \Jj 0 = 0. It will be 
necessary to subtract this cyclic motion therefore from the result obtained by applying 
the condition above. This condition gives 
g/2 
£A„B/» cos nv—\(r Y . , , 
for all values of v. 
Therefore 
