MR. F. W. DYSON" ON" THE POTENTIAL OF AN ANCHOR RING. 
1061 
§ 14. To find the fluted oscillations, it is necessary to determine the kinetic energy 
in the disturbed motion. 
Let $ be the velocity potential of this motion. 
Since the bounding’ surface is given by 
R = a [ 1 + cos nx], 
therefore 
01t* •/Ti.cHo \i n -e/o* \/l ' 
^ c = a(1 + 2 cos nx) + a%/3„ cos Jiy — « 2 {nfS ,,sin ny) [ —, ^ 
at the surface of the ring. 
Now 
0R , 0y sill y 
= cos X and ^ — 
0c 
0C 
11 
Therefore 
0<J) 
0Pt 
= a — c cos X + - («A + «A) cos 7 ix —at sin ny) ^ ' j.“ c 
1 0$ Sill y 
Since 
Therefore 
0R 
A pproximately 
Therefore 
This gives 
a^c ^1+2 = const., 
- + - + Sftft = 0- 
c a 
• / • * 
c f cos X + — j — 5 2AA + ^ («A + «A) cos 7ix 
^ / n • V / 1 0<I> sin y • \ 
a2(nA.sinwx)(^- - ^c j. 
0<1> • / a 
an =-"rsx + 5; 
^ • fR , a Pd 
0) = _ cm 1 - cos X + “ — 
[« ^ 4« a- 
04> • R . 
- = cm sin y, 
Substituting in the last term, we see that it vanishes ; the second term may also 
be omitted, giving 
0^ • / fit \ 
0 J> = — C (cos x + 7y) + t + «A.) cos nx 
at the surface. 
