60 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Now 
when r>c, and 
cl<l) 
=r 
J n 
(I(j) 
' 0 \/+ c- - 2c?’ cos </)) J 0 \/(?■' — sill'" (j>) 
d(j) 
= f 
^(c~ — ?•- sill" (f)) 
when ?’<c. 
Therefore at points in the plane of x, y, further from the axis than the ring, 
v = “|f 
d(j) 
+ 
a~ 
sill" (f) d(f) 
— &c. 
0 v/(w — c^ sill" (f)) 8 J 0 — c“ siiH 
1~.3K . . (2?i - 3)^ (2n - 1) 2fl-» f- mi~“4>d(j} 
^ 23.42 . . . {2n - 2f (2?i)- (2ii + 2) Jo (r^ - 
And at a point in the plane of xy between the axis and the ring’ 
&c. 
M 
TT 
f r 
«2 r- 
d(t) 
1 '0 v/(c“ - r3 siii3 (/>) 
” 8 Jo {c 
:3 _ r3 .siii3 (/))t 
12.33 . . . (2n - 3)3 (2?i - 1) 2«3» |- 
23.43 . . . (2n - 2)3 (2?i)3 (2n + 2) 
r 
J 0 (c" — ?’3 sin3 c^)"'’'^ 
Ac. y. 
The series in its general form must converge at about the same rate at which 
these two particular cases do.' Their convergency is easily discussed. For 
and r is > a + c. 
Thus 
rrr ^ ^ 7j- 
)o (r3 - c3 siir (/,)’'+i ^ ’ 
TTr siid” ^ dy> 
I/O 0*0 
Jn (?’" — c“ sill" 
< 
TT 
Jo (?’" — c3 siir (2c + a) 
< 
TT 
(2c) 
Hence at ail points in the plane of the central circle, beyond the ring, the series is 
more convergent than the series whose general term is 
(- O".; 
2n{2n + 2) 2"c"c” ’ 
i.e., than the series whose term is 
- dd”. 
n (■« + 1) \2cj 
Similarly at points within the ring, in the plane of the central circle, the series is 
more convergent than tlie series whose general term is 
•] 
n (n + 1) \2c — a 
* This consideration of the convergency was inserted at the suggestion of one of the Referees, 
August, 1892. 
