448 
Proceedings of the Royal Society 
negative, and therefore v imaginary ; and for such the solutions in 
terms of o- and the I;, 3£; functions must he used. 
Case II. — Hollow Irrotational Vortex in a fixed Cylindric 
Tube. 
Conditions — 
and 
T = — ; r = 0 when r = a; 
r 
P + p = 0 for the disturbed orbit, r = a + fr R dt 
( 22 ), 
a and a being the radii of the hollow cylindric interior, or free 
boundary, and of the external fixed boundary, and r a the value of r 
where r is approximately equal to a. The condition T = c/r 
simplifies (9) and (14) to 
and by (7) we have 
Hence 
e= - 
1 
m 
dw 
dr 
iw 
, and r = — 
mr 
■ (23); 
dhu 
1 
dw 
i 2 w o 
• (24), 
dr 2 
r 
dr 
- — - - m z w 
r i 
-4( 
icr 
)w . 
(25). 
w = Cl; (mr) ■ 
+• (mr) 
(26); 
and the equation of condition for the fixed boundary (radial velo- 
city zero there) gives 
CV i (ma) + a' i (ma) = 0 . (27). 
To find the other equation of condition we must first find an ex- 
pression for the disturbance from circular figure of the free inner 
boundary. Let for a moment r, 6 be the co-ordinates of one and 
the same particle of fluid. We shall have 
0=f 6dt ; and r—f rdt + r Q , 
where r 0 denotes the radius of the “ mean circle ” of the particle’s 
path. 
