1104 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
Thus, when we are given and we have two equations to determine k and 2 ; 
or given 2 and k, the greatest distance apart of the two rings and their radius at that 
time, we have equations to find and k^. 
One of the quantities, a.,, a, is arbitrary, being the cross-section of the ring, 
when it has a definite radius. 
Let us take 
K 
n 
Also let 
= k\/2 sin 6q and = k\/2, cos Bq. 
Then the equation connecting 2 and Bq is 
sill ^0 n o 
^73- [log 
COS Br, 
I + I log (<^/2 sin ^ 0 )] + 8n - 1+ I log {^2 cos 6'o)] 
1 p sin 2Bq cos 4> B4> 
= log Svi — 1 -h ^ - // --^.(124). 
^y(l — sin 2^0 cos <^) 
cos (f) dip 
/\J (2 + ^2 — 2 cos 
This equation can only be satisfied when B^ is between certain limits /8 and 
( 77 / 2 ) - 
The limiting values of dy give 2 = 00 . 
When Bq is between ^ and {nj2) — yS, the equation gives a real value of 2 ; this 
value becomes smaller as Bq approaches 7r/4. 
We have, therefore, the following theorem. 
If k ^2 sin Bq and k \/2 cos Bq he the radii of two coaxal vortex rings of equal 
strength and volume tvhen the rings are in the same gfonie {Bq being < njA, so that 
k \/2 sin Bq is the radius of the inner ring): then these two rings will continue to thread 
one another in turns, or will seqmrate to an infinite distance according as Bq is > or 
< A ivhere (3 is determined by the equation 
sin /3 
[log 8?i 
V_ 
4 
+ I log (v^2 sin ^)] + 
1 sill 2y3 cos <p d(p 
v/2 Jq \/ (1 — sin 2yS cos 
[log -4 f log (\/2 cos /3)] 
= log — ].(125). 
The following Table gives the values of /3, and thence of 
values of n. 
and k. 2 , for ditferent 
