MR. F. W. DTSO?^ OJs'' THE POTENTIAL OF AN ANCHOR RING. 
59 
§ 6. To find the potential of an anchor ring at any external point. 
When a = 0, that is when the generating circle of the ring is infinitesimal, the 
potential at the point r, 6 is 
]\I H d(j) 
TT J 0 \/-- 2cr sin 6 cos <p} 
When 9=0, and r = z, i.e., for a point on the axis, the expression reduces to 
M/tt tt/R, i.e., to M/R. 
/ d V r dcf) 
\c dcj ]o\/ {?'" + (r — 2cr sin 6 cos 0} 
satisfies Laplace’s ecpiation, and at a point on the axis of the ring becomes 
/ d Y' r (^4> 
[cdc) Jox/( 2 - + C'h 
= (-')" 
.TT ]. 3. 5 . . . {2n — 
5 ■ ■ ■ (211 - 1 ) 
jj2» + l 
1) 
1 
(C2 + 
Now at any point on the axis of the ring 
a a" 
R 8 PC G4PC 102415 
— &c. — 2 y—nr 
P.?d...(2a-3)4(27^-l) 
a~ 
244'-.. . (2«. - 2)~.(2n)~.(2n 4- 2) 15 
) 3 )i+l 
— &c. 
Therefore, at any externaJ ]ioint r, 9, 
rr_ <^<t> 
TT [Jo y(i'~ + C“ — 2crsi 
(f d 
“T o 
f 
J n 
dcj) 
d \3 H 
sin 9 cos (f)) 8 c dc J o s/(r~ + c- — 2cr sin 6 cos cj)) 
+ &c. 
H d 
)o\/0-- +c2-l 
J_ (_ I V'+l - . , , 
^ ’ 2/1 + 2 2-.4-.(3“... (2?i)® \cdc) J o \/(r“ + c-— 2cr sin cos 
192 \c del Jo \/ (i'~ + C' — 2c?’ sin 9 cos 0) 
2a~“ 1. 3. 5 ... (2n — 3) f d \" p" d(i> 
+ &c. 
For this series is finite at all external points, satisfies Laplace’s equation, vanishes 
at infinity, and agrees with the value of V on the axis. The series is very convergent, 
as is seen by the next paragraph. 
§ 7. The integral 
/ d y r_ d^ _ 
\c dcj Joy/ + c^ — 2cr sin 9 cos cj)) 
takes a simple form at points in the plane of the central circle. For, putting ^ = 7r/2, 
it becomes 
/ d Y' p_ d^ _ 
\cdcj Jo \/{i'~ + c- — 2crcoscj>) 
I 2 
