1076 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
Therefore 
Put 
Then 
Const. = ^ + 2 log 2 — log \/a^ — W. 
# = 
y — cosh IV = cosh [u + w). 
cosh w (sinh w — cosh w) i i i -is i 
b = - - -log (cosh IV + sinh id) p ^ 
1 1 
ilog-9 
= - 
\_ p — Zw _ JL 
log 
cl 
Taking the real part of this, we find for the first term of the potential 
16c u 
i log — — T) — i e cos 2v 
cl ^ 
a*' 
16 c 2 . 
Calling the operator 
1 I 1.1 I p 
Since 
Therefore 
/(D. D')t«’ = ®/(D, D')« + DtiY„ + D'®^«+ ka. 
/(D, D')(log|-l)^^ 
= ^^/(D,D')(log|-l) 
. / 1 Pt COS ;\;\ 0/ /, 8c 
+ (2-.-^)aDrsE-i 
+ &C. 
It is easily shown that 
S/ /i_8« 
0D 
log ^ — 1 j = (e gQg 2,')) — 3e“" cos v) + 
12 d 
Fc 
Pietaining only the terms of the highest order 
r A 16c 
2c 
a- 
(46a), 
(47). 
.z’/T-v 1/1 8c \Rcosyl co.sh ?{ cos v f, , 16c w , . , 
/(D,D)((log-- =- 5 ;-jllog— J_4c-2.c032t, 
d 2- 
^ I' 
+ -r—A (<3 cos Sv — 3e “ cos 
24cd '• 
cosh u — -j;^g cosh u e — g-- e "/ cos t’ 
+ ( — iV cosh u e + —r^ e~^") cos 3v }- . . 
(47a). 
