MR. V. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
/ 0 
This is evaluated by picking out the coefficients of cos y;, &c., in the expansions of 
J^, Jg, &c., given in § 3 (C), and 
z=\Trau s A;i ( — 
X + 1 
o- + 
3a, -(- 0 
cr3 +Ag cr- 
20X + 17 
32 
0-3 + 
18X + 29 . 
O' 
256 
-f" A 3 -f 0-3 -h A^ cr'^ 
Mid 
1 - 
12X + 5 . 144X2 + 24X + 57 
16 
0 -^ + 
512 
cr 
§ 13. Ring rotating about a Line through its Centre in the Plane of its Central 
Circle. — The velocity potential 4> may be taken to be of the form 
Ai + A^a^ I cos (b. 
'■dz " dz J ^ 
The coastants A^ Ao, &c., are determined by the boundary condition 
or 
y = — cwo sin ^ cos (f), 
ds ~ ~ ^ 
dll 
d<P 
Differentiating the formulas (D) of § 3, 
d /dJ, 
ds \ dz I 
If. / 12/-1 . 3 ■ o ^ - 
. + 1-0 5 • , \ 3 
+ Sin 2x - A «in 4x 1 s3 
, /m - 3 . , 1080^ + ^ „ 1 O' • - \ .1 
+ ' 2048 ^ ^ 
I (5) {2 sin 2x + (f sin x + i sin 3x)s 
/ 35 Z — 0, ‘X * rt 1 
+ ( g 4 sin X - tIs sin 3x - yls sm 5x ) + . • r 
d 
ds \ dz 
d /dP 
^ {6 sin 3x + (5 sin 2x + sin 4x) s 
Ct a 
+ (¥ sin X + TF sin 3x + -h sin 5x) + . . .} 
ds \ dz 
d/dj, 
ds \ dz 
J-. (24 sin 4x + sin 3x + | sin 5x) s + 
C“S' 
> 
•} 
= ^ 120 sin 5x+ . . 
L 2 
