SMALL VIBRATIONS OF A HOLLOW VORTEX. 
179 
cos nv 
I o (C—c) 
C* cos nv 
idv- 
2\/2 5Q K 
Jo(C-c) 5 
O0 f 7r cos 
S 2 77:—-1 = 
Jo(C-c) 5 
S du 
4y/2 5 /I 5Q, 
3S [5 m \S c^m 
W2/d?Qn_C 5Q* 
3 \ 5m 3 S du 
\/2 f, 2 8C d Q n 
-J-1 (Ira 3 — 1 )Q„ — — — 
Hence dropping the dashes, and u denoting the value of u along the tore 
i(A 0 +tA 1 )P 1 (Q 0 -^T 0 ')+|A 0 P 0 ('3Q 1 +^T 1 ')+2 1 B„{(4« a -l)Q„+ 8 |T„ 
A u having the values given above, and the values given in 7. 
y. The value of y is given by 
f 2ir ilX! S\[r 2 
2 y/xx^=^j o 
tt • i i 
Here iio is constant and —0, also 
Jo p OM 
Hence 
#»= iV(/-Xa»)=*V~ W) 
tv 
y=vrpf (p 3 —Xa 2 ) l ~dv 
' crynJ 0 p 5 w 
o J 0 
Further j — is the flow along a closed curve threading the aperture and 
is 
therefore the cyclic constant y. Therefore 
7= 
7 t\ 
^ [ 
I 
O 
J 
1 o J 
The last integral may be expressed as in the analogous case for yd. 
Section III .—Steady motion of hollow vortex. 
9. The form of a hollow vortex and its motion are conditioned by the fact that the 
velocity of the fluid relatively to the hollow, when the motion is steady, must be 
constant over the whole surface of the tore. When the section is small compared 
2 A 2 
