MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
85 
If the cross-section of the ring is very small 
A _ 
.a.1 — 
KC 
iTT 
and 
4. = const + + (;¥ * + sin x + s' 
/6^ + 7 ^ 15 -f- 7 , . . . 
I--I-1 sin 2y 
oZ -f 4 ^ . 35^ 20-fV 4, • ^ , 
-f sin 3x + —sin dy -f .. 
But in this case <I> is proportional to fl the solid angle subtended by the ring at 
any point. 
Hence the solid angle subtended by a circular ring at a point near it is 
T 1 f^+- . 3^ + 2 
21 ^-X- 64 
3\ • + 7 0 , 15/ + 7 A . , 1 
s^j sin X - (^^ 0 ^ s-' + - 3 - 3 J- s^j sm 2 x - &c. | 
§ 17. The kinetic energy of the fluid has been calculated for the different motions of 
the ring. To find the kinetic energy of the solid ring, its moments of inertia round 
the axis, and round a perpendicular to the a,xis through its centre must be found 
Let p be the density of the ring, and M' be the mass of the ring. 
The moment of inertia round the axis is 
[ [ 2TTp' (c — R cos x)^ R f^R c^x 
JoJo 
= IttV |cR2)R(7R 
Jo 
= M' (c^ + I «2). 
The moment of inertia round a perpendicular axis is 
0''27rp' (c - R cos x) ^ Jjs ^ 1,^3 
= M' (I -I-1 «=) + M'. 
= M' + f . 
§ 18. Therefore, p being the density of the fluid, M and M' the masses of the fluid 
displaced by the ring, and of the ring, respectively, the kinetic energy is given by 
2T = P (n2 -f yS) + R^(;3 + A 
