1084 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
will be shown to be of the order 
—— r.e., - 
c”- a 
c, 
We have assumed to be of the second order, ^3 of the third, &c. 
/Ci 
the above formula, all terms as far a-s (“j have been retained. 
S 33. Let 
Then 
Let 
Then 
Or, 
=r 
j 0 
c cos ^ d(f) 
y /— 2z!s'e cos ^ + z'-) 
cm , ^ 
cm ccic^ ^ 
U = I. c. 
cP , cP \ [cPl , 2 cll , cPl 
dc- dz'~ ' ^ ^ ^ r. rln'^ ,1 
V^U = 3 f = 
dc 
V^'U = 3 I- V2I 
dc 
clc~ c dc dz' 
dc \ c dcj 
vou = 3.1 y V2 5r 
clc \c dc 
Hence 
3.1y|-V2y--4(-^U> 
dc \ c dc c dc \c dc, 
V-U = 3.1(-l)...(-2«+5)|(i|y‘-‘l 
Therefore, in 
(65a). 
§ 34. Tlierefore at a point nr', z, cfj', the stream-function is given by 
