1046 
MU. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Let 
Then 
Therefore 
U„ = e'"* cos n^j. 
^ "A . i _ cos n — Iv. 
c7p- djc 
TT + COS — Iv 
U„ + 1 _ an + 1^2 -in- 1)2 ’ 
= ^ g -0 + i)p eos {n — l)^. 
This gives 
+ 2 
f7p2 
. f7“U„,9 3 \on / . 2 % + 1 1 
+ + |— cos (^^ — 2)x -1-cos j' 
Therefore 
U«+3 = \\ cos ?^x + #2 cos (?i — 2) X 
16 {71 + 1) 
In the same way we find 
U^ + s — 
a 
, {n + 3) p 
, (271 + 1) {2n + 3) 
^ (2?i + 2) (2?i + 4) 
1.3.5 
cos {n + l)x + 2 - 
1 .3 2 ?i + 1 
2 .4 2 % + 2 
-cos(n- l)x 
+ : 
2.4.6 
cos (?^ — 3) X f- 
This suggests the form of 'U„+p, which is easily verified. 
cos (w fi-p — 2) X 
1 2^1 (2^ 2yTT7(2iirf2^^^) + P “ 4) X 
^ 1.3.5 72».-l-lL..72n,4-2-}i_ 
n — /fiV gO^+iP)P J1 + 1). ■ ■ (2ii + 2p - 3) 
— [ 2 )^ 12 + 2)... (2n + 2p - 2) 
— 1 1.3 (2n + 1)... (2?i + 2^ — 5) 
1 ^ ( 2 ^ 2yTT7(2iMr2p^^) 
0)-l)(p-2) 1.3.5 (2«+l)...(271 + 2^-7) 
2 ! 
o ^ ^m /.T o ... COS (n+» —6)x + ...l (2). 
2.4.6 (271 + 2)...(27i + 2/)—6) ' ^ J ' ^ 
This formula for holds whether ^ be > or < n. The number of terms in 
IS 2^' 
§ 3. The series + - U,,+i + U^+o + ... is a solution of Laplace’s equation. It 
C C'^ 
is convergent at all internal points. 
a\P 
For is < e<»+2^)p(l + l)p-\ 
Thus the series U« +-U„+i . . . converges more rapidly than the geometrical pro- 
1 + - + ? + &C. 
C C" 
gression 
