SMALL VIBRATIONS OF A HOLLOW VORTEX. 
183 
To the same order 
Therefore 
—y= —(L—2) 
7TO ' ' 
by (15) 
V= 
47T(X 
(L-i) 
( 21 ) 
The principal term in U is found by equating it to the principal terms in a, i.e., 
-p, k~ l 
2^ 0 2tt 0 4 L—2 
and is therefore independent of the velocity of translation, as ought to be the case, 
since the latter depends on the difference of the cyclic tangential velocities inside and 
outside the tore. Substituting for i Jj 0 
U - ---- 
0 4 irak 
Now, for steady motion, the equation of pressure gives at the surface of the hollow, 
if II and p be the pressure at an infinite distance, and the density of the fluid 
respectively 
Hence U must be the same for hollows of all sizes, and consequently ah constant for 
all the steady motions of the same vortex. When the hollow is small this is approxi¬ 
mately the same as saying that the radius of the cross section is constant. The 
corresponding theorem for a solid ring is of course that the volume is constant. 
10. For the second approximation we need to determine the stream function when 
the cross section of the ring is not an exact circle. The following investigation is 
slightly more general than is necessary for our present purposes. 
Let k be the value of k for the mean section, and let the section be given by 
K=k-\-t( M,„ cos ra;+N„ sin say 
where M„, N ;i are small quantities with respect to k. When the tore is at rest with 
fluid streaming past it, the stream function is 
xjj= —xx]j 0 
S 2 
(C—c) 2 
2-v/r 0 \/2 ^ A Pi„ 
where the A u have the values given in (19). 
