MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
104.5 
, cm 
cim 
clz 
~2 
= 0 
tlx^ 
cm 
(jm 
I ^ ^ n-ir\ 
+ 
«+2 
clx’^- 
+ 
dz^ 
dm^^ 
«+2 
dz^ 
= X 
= X 
d^JJn 
dx^ 
+ '~d^ + ^ 
r^+i 
Let 
then 
therefore 
Take 
then 
dx^ ' dz^ 
&c., &c. 
U,, = {x -\- izY, 
I 
"*■ (/rt; 
f?a;2 
U«+i = + C (a? + izY'^K 
C = 0, 
U„+i = i (a? — iz) (x + iz)\ 
From this U „+2 might be found, and so on. 
§ 2. It is, however, more convenient to transform the equations to polar coordinates. 
Let 
X = ae'’ cos ^ sin X’ 
Fcr points inside the ring p varies from — qo to 0, and x from 0 to 27r. 
Now, 
dU 1 / dU . dU^ 
= - e (cos X 
dx a 
dU 
dz 
i, - X 
1 — / ■ , V <*u\ 
= - e sin Y — -4- cos A — , 
cc \ ^ dp dx! 
and 
d^U , d^u _ 1 _ 2 „ (cm d^u 
dx^ dz^ ~ cd ® " Vdp2 dx^ 
Substituting in the above equations, we find 
d"U„ , d^U„ _ 
dp^ dx^ - 
d^u„,^ . d2u„+, „ f (cim, , cm,, , dUA . du„ 
dp 
&c. &c. 
d^U„+2 , d2U„+2 _ \ f d-Vn + i + 1 , dU.^ + ^ 
dp 
- s^x 
dUn + l 
} 
• • (!)• 
