MR, F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
91 
§ 22 . The potential at any point external to the ring whose polar coordinates are 
r, 0, is found by writing 
1 (•" c/(f) 
ttJ Q\/('r^ + — cr sin d cos 0 ) 
instead of 1 /R, l/R'^, &c., as in Section II., § 5. 
Instead of l/R, 1 /R^, 1/R®, &c., we write 
&c., 
h R I 3 
I, 
1 = 
7r ’ TT ’ Stt’ S.Stt ’ 3.5.77r 
Therefore, the potential at any point outside the ring is given by 
V 
where 
2 ^,, 2 , = All] + I 2 + AgaV I 3 + &c., 
A.= .+f 
K=-\-\<yh 
^3 = ~ f^.2 + (i + 4T ~ 8 CT/dg 
"^4 — 'H ~ 2Q.)4 “ 8 + 2 ^3 
A 
_ / _ f7 * _ 0-2^2 ^ 1^1 
^ ~ ^ I 2’^ 256 32 8 32 / 
Using the expansions given in § 3 (A), the potential at a jjoint R, y near the surface 
of the ring is given by 
A 2/HI 2 . 108/ + 27 , /2/ + 3 . 12/+r) 2 \ , a o 36^ + 51 
, = A, O + 2 + —5‘ + Ajor --- + —s" + AjO- — 
2 /„==A,^/ + 2 + —+ ^ , 128 32 
, ,,//+! , 9Z + 3 , /I , 12Z + 11 , ,9 
-h cos X ^ Aj ( 6' + 6' ) d- A^O- ( ^ H-5 ) + AgO- 
64 / ' Vs ' 32 -y ' - 3 - 45 
, ^ /3/4 2 o , 20/+4d aA . a . ^2Z + 9 A 
-h cos 2x IAi s~ + « ) + A^o- (^5 + ' ^4 ” ^7 
+ AgA' + I j + A^.cr -2 1 
-h cos 3 x j Ai - + Aoo- 3% s + Agcr^ ~ + A^c^® ^ 
+ cos 4 x j A^ ^ J 208 ^ iIa + Agcr^ 3-2 + A^cr^ — + A, 
where as before, I stands for log ( 8 c/R) — 2, s for R/c, and cr for ajc . 
N 2 
4. ® 
5^^ 
