SMALL VIBRATIONS OF A HOLLOW VORTEX. 
175 
To find the point where the two sets meet on the tore, we notice that the stream line 
there goes to infinity, and its value is the same as for a point on the axis, it is, in fact, 
a part of the same stream line. For this \Jj=0 ; hence the point on the tore, where 
this stream line meets it, is given by the value of v, which satisfies the equation 
(1— X)T 0 + 2Sr^l+ 4? ^^^T„ cos nv— 0 
where T„ are the values of T„ when u—u. 
It is clear that when k is very small, cos v must be negative, that is v > \tt, or that 
the point of division must lie inside a tangent from the centre to the tore. 
7. Combined translation and cyclic motion .—The expressions just obtained enable 
us to determine the amount of fluid carried forward bodily with the ring. Let x 
denote the ratio cdV/2xjj 0 ; then the stream function for the combined motion is 
where 
, 2f oX /2 ^ R„ 
■' , =i7(c^) 2A *B', cos nv 
A 0 — {1 + (1 — } T 0 
Kn ~ 2 {O 4?i 2 -l } T *■ 
This is the stream function when the fluid is at rest at an infinite distance. To 
find the portion carried forward, impress on the whole system a velocity equal and 
opposite to V; the problem then is to determine the portion of fluid which remains 
circulating round the ring at rest, without streaming away. The stream function for 
the new motion is 
The portion remaining with the tore lies inside the surface given by putting y equal 
to a certain constant, which we proceed to determine. 
This portion may either be ring-shaped or not. The limiting case between the two 
is when the velocity at the centre of the tore is zero. The value of x for this case we 
shall call the critical value of x. It is given by 
V= 
1 b\[s du 
_p bit dn 
U=0,v=n 
_St aK /2 1 / 
7T» a U '“°s| 
=>(-«**- i)& 
V 1 
>' A Y+ 72 2( ->* A >' 3 - i) ^ 
or 
