MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
1093 
§ 39. The equations of Motion. 
Let the ring be moving forward with velocity z-^ 
Let its radius increase with velocity Cj^. The radius of the cross-section will change, 
but since a-^ Cy = const., 
«i = - i 7 Cl, 
A 
and therefore being small), is negligible compared with e^. 
The normal velocity of any point on the ring’s surface is 
2 i sm X — Cl cos X — oi^. 
But the resolved part of the velocity along the normal to the ring is 
Therefore 
(Mr 
ds 
xn d-m ds m dz ds 
= cTT (zi sin X ~ Cl cos x — «i) 
(91) 
at the surface of the rino-. 
Therefore at the surface of the ring Cp 
= I (zi sin X “ Cl cos x “ cii) ds 
= j" (zi sin X ~ Cl cos x — ^i) (ci — r^i cos x) «i f^x 
= — «i Cl Zi cos X — Cl Cl sin x + terms of higher orders . . (92). 
Comparing this with the value already found for at the surface of the ring, we 
obtain 
Cl Zi — 
riu dl 
Wg fZIi3 
cq V ^ TT dc, + TT f/q + 
1 
and — Cl Cl = 
If we write 
then this equation may be written 
■wi.2 (ZIi2 ^ 
TT fZ^i TT (7~i 
+ 
(93) 
S » Ii3 = U, 
TT 
m,CiZ, = 7log(^‘-p + ;7 
and 
27r 
Sq 
— /^l Cl Cl — 
clU 
~dz. 
(94). 
Similar equations hold for each of the other rings. 
