448 
Proceedings of the Royal Society 
negative, and therefore v imaginary ; and for such the solutions in 
terms of <r and the I*, f; 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 + ff%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 f 
where r is approximately equal to a. The condition T — c/r 
simplifies (9) and (14) to 
1 dw iw 
p= ~ — , and r = — 
6 m dr mr 
■ (23); 
d 2 w , 1 'dw i 2 w 9 
dr 2 r dr r 2 
• (24), 
and by (7) 
we have w = — (n - w . 
m \ r 2 ) 
(25). 
Hence 
w = Cli (mr) + (mr) 
(26); 
and the equation of condition for the fixed boundary (radial velo- 
city zero there) gives 
CP; (ma) + d3E'< (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 0 , 
where r 0 denotes the radius of the “ piean circle ” of the particle’s 
path. 
