MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
77 
§ 14. —Stream Line Function for a Ring moving parallel to its Axis, and for the 
Cyclic Motion through the Ring .—Let the velocity of the ring be iv, and the circula¬ 
tion be K. 
Let xjf be the stream-line function. 
Then 
Cpyjr 1 d\Jr 
dA dxs^ m duj 
at all points of the fluid. 
At the surface of the ring 
xp — = const., 
or 
xjj = Ak (B — cr cos y + ^ cos 2^^ 
We may therefore assume xp to be of the form 
{A,J -f A^a^JgCT -f Aga^JgCT -f , . .]. 
Using the values of Joct, &c., given in § 4 (E), 
The equations giving the constants A^, Aj, &c., are 
- 
16 
^. + 1 , 3A. -f- 5 
2048 
+ Ao 
cr 
o /2X - 3 , 20X + 47 o' 
+ 
128 
+ -^3' 
52\ - 57 
= A/c -f Biyc~ 
d- 
64 
cr'^ +Ajl- 
20X + 17 o 
12\ + 17 . 
768 
-i- 
cr" 
8X -h 1 o 
-h Ag ^ cr” = — IVC^ 
64 
O’" + ^3 (1 — I O'") + 2 A 40 -” = 
1 192 . 32 4 ~ ^ 
. 15X — 8 A, A., A, . 
“ “ 128 “ 32 ~ 2“ + = 0- 
Another equation is found by using the fact that the circulation is k. 
Integrating round a circle concentric with the crosss-ection (radius < c), we have 
