84 
MR. F. W. DYSOR ON THE POTENTIAL OF AN ANCHOR RING. 
(Jj = const + ^ X 
TT 
\ [I 2 .3Z + 2 . I + 7 o , 15^ + 7 ^ . 
+ 7 K’T’ " + “Sr * ) X + (-^2^ *- + **) 2X 
5/ + 4 . 35^ + 20yV 4 • , 
, A.cr^ \(^ , 20/+ 37 . , is , 16/+ 21 ^ 
+ ^ [y + ^^ + (i + —^ ^ 
1024 
X 
4 ' 64 
+ ^ s- sin 3x 4- tIs sin 4x 
+ {^‘T s sin X + (t + I -s-^) sin 2 x + i 5 sin 3x + 3 % sin 4x] 
A„cr' 
+ 
C“S' 
AV« 
“4 '!> sin 2 x + 2 sin 3x + ^ sin 4x 
I -^5^^ i • A 
+ 77 N- 4 sin 4x. 
c*s” 
This holds only at points not far from the surface of the ring. 
At tlie surface of the ring s = cr. Substituting the values of the constants given 
in § 14, we find that here 
d> = 
iir 
x + 
2Y + 3 
cr 
8X" + 28A + 11 
64 
cr’M sinx + 
16X +15 „ . 
32 
cr' sm 2 
X 
57X + 49 . I 
H-^^^cr^sm3x + &c. j-. 
This gives the impulsive ] 3 ressure at any point on the surface of the ring. 
The momentum of the cyclic motion may be deduced. 
It 
= 77 (c — ffl)^ Kp — \ \ sin X dS 
= TTKp (c — aY — 27ra O sin ^ {c — a cos x) dx 
= TTKp (c — a)' — '^^ 27 rac j (sin X “ 9 sin 2 x) 
rZrr 
. 2X + 3 8 X^ + 26X +11 ol . 16X + 15 „ . ^ \ 7 
X + i —9 a- H---- o-' Sin X +-^3-cr- sin 2 x) dx 
32 
= TTKpc^ ^1 — 2(7 d- cr- + ( 2cr — — j — A- ( - + 
o /2X + 3 , 8X2 + 26X + 11 
64 
cr'^H- 
16X + 15 
64 
TTKpC- ] 1 — (A. + 1 ) cr- 
4X2 + 5X, - 2 
cr' 
which agrees with the result already obtained. 
O 
