82 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
where t/ijj denotes the value of ifj at the surface of the ring and ifjj^ at the axis. This 
result applies to the cyclic motion through a circular ring of any cross-section. 
We have taken 1 /;,^ to be zero, and called x{/^, A : so that we have T = npK A. 
The Linear Momentum of the Cyclic Motion is parallel to the axis of the ring and 
= p III — to cZtjy dz dff ): 
This integral 
the integral being taken all over the fluid. 
= 47rp I dz 1^^ drn 
= 47rp [ + V'E' — '/'a) dz -f 47rp [ {xp^ — xpf) dz. 
Jo j ft 
We have taken xpj^ = 0. 
xp is constant on the ring, so that 
Therefore, the resultant linear momentum 
= 47rp f xp^ dz, 
J 0 
where ip^ means the value of xp at an Infinite distance from the axis. 
Consider first the part of xp^ arising from the term A^J^nT. 
This is the limit of 
w cos <p dtp) 
10 ‘Z (z" + d — 2(7rc cos cp + 
A.f 
where ct is infinite. 
This 
= Af 
J 0 
, niC COS (/> I , 7 , 
. TT cm“ 
= A,;: 
2 + 
.9X5 ' 
