MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1065 
Therefore 
Mx - 4- - f) = 0. 
(Co + xy -lir (Co + xf \ ^ (ci^ + y) ^ > 
Therefore 
or 
or 
«n"Cn~ ( lOQ- — 4 , 
X-y _.V ^_7^vr.o-Co 
0 '"0 \ ^ 4 / O/ 9-7r r/ Jr... /_ 8 (Co + x) 
(co + xy 
Itt (cq + x)~ 
±A^og 
-1=0, 
% +1/ 
+ 3y '^(log -i)x- 2y 5^(log p-i)x+fy'^x = 0, 
^0^ \ ^^0 
.T -f- y ^2 (log ^ — 0, 
C^7), 
giving for the time of a complete oscillation 
-•^6) / 
CL 
The oscillations might be found more approximately, but this hardly seems worth 
while, as the ring will be proved unstable for disturbances of a different kind. 
§ 16, The effect of long headed ivaves. 
As before, let U be the exhaustion of potential energy of the ring in its disturbed 
state. In this case tire ring is disturbed so that its central circle remains an exact 
circle of radius c, the cross-section is always a circle, but the radius of this circle 
varies with the azimuth. Let it be given by 
p = a{l + S (a/i sin iff + A* cos ncf))]. 
Let Vj be the potential inside ; Vg, outside. 
Then 
U = 1 |V, dm 
= i f f I Vi (c — It cos x) If cZE, dx dcf). 
J 0 j 0.' 0 
If Vi be expressed in the form 
-i- Vi cos X + V 3 cos 2x + . . 
the only terms we need are Vg and Vi, 
Also, the only terms of the second order in the small quantities a and /3, which 
need be retained are those independent of cf), as the others vanish on integration. 
MDCCCXCIII.—A. 6 U 
