10G8 MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
It is important to notice that at the surface of the ring differentiation with 
respect to c lowers the order of the terms once in ajc. Therefore the operation 
— • y raises the order once in ajc. Provided that n is not greater than ^{cja), that 
is, when the waves are long, the value of Vq is given by § 17 correctly to the 
order 
V/e shall, however, simplify the work by rejecting all but the most important 
terms, though it would be quite easy to retain terms of higher orders. 
§ 19. Let the point w', z, (f)' be near the ring, and with the notation used before 
be the point R.y. 
The most important term in [ - — 
' 0 
cos n4>d(b , . r. , 1 P 1 r cos nd>d(f> 
The integral ^- , , o "—being of the form ■ , , 
° J “ c cos (^ + c- + *") ■ ° A/(2uy c) J 
. , 8c 
- 2^'c cos (f) + c^ + p2) p 
IS 
1 1 J 160+1) 2^2 / 
^(2,.'c) v/a 4-1 v'{2»'c) I + 
>^(2cj'c) Jov^f? — cos (f)) 
1 
2n ~ ly 
(J. J, Thomson, ‘Motion of Vortex-rings,’ p. 26) 
8c 
where 
= log— - 
/(?l) — 1 + A + • • • + 
2?i - 1 
Writing </> for the azimuth, instead of (f)', we have, therefore, that near the ring 
outside 
Vo = 2,r«»|l + S^-^’} log I 
-f 4TTa^ 't {dn sin + A, cos ??</)) j log y - 2/(w) I . . (30). 
Inside the ring, therefore. 
v, = 2™qiog®-' + i(i-^"' 
+ iira^ S (a,, sin jx/i + (i„ cos jlog’^^ — %f (n)| 
+ c. 
■ ( 31 ). 
where C is a constant of the second order in a,;, kc. 
