72 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
At the surface of the ring s a = ajc : and we have the following equations to 
determine the constants :— 
A I 1 "t 1 0 18X + 3 
1-” 
I (A + 3 
i “ “ 32~ 
- A- 
256 
o-A + 2 A 3 +A 3 5o-2= 0 
j 
360X + 183 
127i + 1 
64 
cr"^) -j- Ao = <^CW 
+ A,(i - + A3(6 + H cr=) + = 0 
2048 
~ btAi + Ag + 24 A 4 = 0 
~ i^^Ai — Tfs-Ag + wg Ag + IA4 + 12OA5 = 0 
These equations give 
3 — 
Thus 
Ai 
Ag 
A 
A4 = 
Ag- 
$ = 
^ . 4X + 1 , , 16X2 + 8X 
^ ^ ^ 
12X + 1 , , 
cT^a-civ. 
1 , 360X + 57f o 
1 H--- cr\. I crcu'. 
I, 
O’" !-a^c«’. 
J 
64 
1 
384 
5 
oil 
64 
ci^ciD. 
a^cw. 
0 I A 4x + 1 ^ 
Cl-'CIV *1 ( 1 + — CT' . 
dlj 12X + I .TO din . ] 
The kinetic energy is given by the equation 
2T = 
d) — c/S 
an 
-.ff 
= — pry 277 (c — a cos y) sin y d) ac/y 
J 0 
=: — 27TpaCW 
ac 
sin Y — - SI 
J 0 
X - dsm 2x)dir/x 
o O r A /i I ^ « Ox + 9 n\ 
= “^-P7. Ai(^ + WW - 'asiT J 
o- ] + A 0 O-" d + 
12X + 1 .T 
64 
C7~^ + AgO-^lj 
A 7 " I 1 I ”t 1 9 I ^ + 8 X — 23 
= Mry-" ^ 1 + O’ H-773- o-^ + • • 
16 
512 
where M is the mass of the fluid displaced. 
