MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
lODl 
Section V. 
The motio 7 i of any number offne Vortex rings on the same axis. 
§38. The stream-line function. 
The cross-sections of the rings are approximately circles. 
In each ring, let the vorticity be constant. Then <y/t!7 = constant, over each ring. 
z 
I;et oj = -^Mgcr, &c., in the several rings. 
The stream-function SF satisfies 
1 
fZ'F 
dz^ 
fZci'^ 
zas 
tZcj 
+ 
fZ2^ 
1 
fZ^ 
did 
djyr 
CJ 
dm 
fZ^^ 
+ 
1 
d'ir 
dz^ 
(Zw“ 
rfy 
dxds 
+ = 0, inside the ring ; 
+ Mara” = 0, inside the ring Co, &c.; 
while and d'^jdn are continuous every’where. 
The value of at any point ra', z, outside the rings is given by 
'T'' = ra" S 
Ttt ]]]\/ {^(f — zT + cos (j) + 
cos (j) dz dv3 d(j) 
2 1 
; 
( 86 ), 
where the integrals are taken throughout the volume of each ring. 
Consider the integral over the ring whose mean radius is c^, and cross-section Wj, 
and whose central circle is distant from the plane of xy. 
Let 
^ = Cl -/| 
z= Zy-\- h] 
6 z 2 
