1072 
J*JR. F. W. DYSON ON THE POTENTIAL OF AN AECHOR RING. 
Write 
sill n(f>' + cos nf' 
sin n(ji' + 8/; cos ncp' 
Then 
r a-/cp . <:P 
Y,, = wa?\ 1 + 8 
+ ..0 |i -f 
O / 70 
ct'' / fr 
/ ^ w 1 p“ CCOS(f}(h 
^ dz' 'J‘ ‘ • J Jo v"{c7'2 + (;2-2cct' 
rfc 
c cos <p d(f) 
cos ^ T" s"} 
+ mr|l + gt7J, + rf,-jj+ • •• 
f_t vt 'I (1 — cos n(f)}d(f> 
^ ^ ^ f^j Jo s/W'^ + c"--2ci^' cos 0+^2} 
Let the point cr', 2 ', (f)' be on the surface of the ring. 
Let 
ot' = c — il' cos X z' = E,' sin 
X 
— c — a cos xi — r 
= « sin xi + t- 
lletaining only the most important terms, 
V„ = 2to’- ( 1 _ f' a + . . . ) jlog + &C. 
„ on 8(c-r) , r r, > 
=::: 27r«-' ^ log--j- ~/(n) 
= 2™= ] log f [/(«) - 1] - g 
= -j log — + S [y (^0 ~ 1] "h A< cos ncf)') 
CC 
■<n~ + /3,r 
. (39). 
§ 22. Inside the ring the potential is V;-, where 
J L.,.. ^ _L 1 
a 
rd 
\i = 27rrt® jlog — + ^ (1 — —) + S [/(n) — 1] (a„sin cos — S 
^ii~ + ^ii‘ 
where is the distance from the curve of centres. 
The exhaustion of potential energy 
U = 
= ijv 
; dm. 
U 
TTCd 
\i 
iira 
- (c — X — dx dz d(f)', 
thus 
