MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
87 
Substituting, we see that the motion will be stable for any value of 12 if 
w + —J [w + -— , ) > 0. 
TTfl"/ \ 27r«" (p + p)/ 
Therefore the motion is stable for all positive values of iv, and for all negative 
values numerically less than 
KC 
or numerically greater than 
Tra^ 2 (p + p') ’ 
KC 
o 
TTrt” 
The motion is stable for values of w between these values if 
^ _ 2p (p + p'f / / KJ^ _ p _ 
p'2 (p + 2p') \ TTcr) \ TTcr 2 (p + p') 
The greatest value the right-hand side can have is 
^ p (3p + 2p') 
TT^ p'3 
Therefore the motion is always stable if 
^ / 
n> — V 
Tra- V 
P (3p + 2p0 
'2 
Another possible steady motion of the ring is for it to move round in a circle, as if 
it were a rigid body attached to an axis in the plane of the ring. Mr. Basset shows 
that if T be the time of a complete oscillation, and r the radius of the circle described 
by the centre of the ring, then r = T/r 7 r. Z/R. When the ring is very fine this 
becomes 
T CK 
r = 
IGtt" p + p ' 
Section V.— Annular Form of Rotating Fluid. 
§ 20 . In order that a surface may be a possible figure of equilibrium of rotating 
fluid, it is necessary that V -f- sin^ 6 should be constant over the boundary. 
Let us assume that p = a (1 + /3^ cos y + A cos 2^ + ySg cos 3^ + . . •) is the 
equation to the cross-seetion of the annulus, and that /3j, ^ 3 , &c., are small quantities. 
