MR. F. W. DYSOR ON THE POTENTIAL OF AN ANCHOR RING. 
1083 
Part (2) 
= \ e-«vco3(x-a) 2 ^ cos 3v + cos 4;^) 
Jo 
r27r 
= a® f ^ 0 ^ cos 2a cos 2x(j + A cos 3a cos 3i/> + A cos 4a cos Axp] dxfj 
J 0 
= rra, 
4: I 
,/32 
4 \dc^ 
1. 
dz'^ 
1 + 
«2V“ 
12 
+ 
— ttw 
12 \ dd‘ 
/Se fd^ 
cf d d? 
dc d^^ 
1 + 
fPV^ 
Te 
+ • • 
d^^ 
dJdF~ 
jP 
(63). 
Part (3) is easily found to be 
TTO^-^ + iT Tra^yS, 
„ / d'*’ 
^ d- d^ , # 
6 —— + 
4!\dc‘^ dc^ dz'^ dz'* 
(04). 
At any external point the stream-function is, therefore, the elfect of the operation 
1 ^ ¥. 1 ^ +•••!■+ — ^ 1 + TIT + YqT + • • • 
4 1 dc- dz'^ 
fdV“ , fdV^ 
IF Fsi 
, yS, / d^ ^ d d- \ 
yO ' 
12 Idc® ^ rfc ds'2 j (^ + 16 + ’ ■ ■ 
/3,. , 5^2\ /d-^ 
+ Tra® (192 + 16(4) (^ fZc 2 d^ dz'^ 
on 
r-’^ c( 
Jo v/— Sct' 
cos (j) dcf) 
CGOS<p + c- + z"^) 
. . . . (65) 
32. Now at the surface of the ring 
[ 
cos 6 d(b . ^ 1 c” 
■„ —s - :r is ol the order 5 
0 y /(rz ^ — 2oi c cos ^ + z!^) a 
d”' c® cos (bdd) ‘ 1 . c- 
w”— .. ^ IS ol the Order a "—-7 “ ’ 
fZc” Jo ^(m “ — 2m c COS (f) -i- c- + 2^^) rt""''! a 
cat" 
d2 cP \» 
dd^ 
c? cos (f) d4) 
0 — 257'c cos 6 4- c" + z'^) 
6 Y 2 
