1054 
and 
.Zo„ 
27ra^„ 
MR. F. W. DYSOX OX THE POTEXTIAL OF AX AXCHOR RIXG. 
ft Y' cos 
R 
+ .^ 
] /aV-^ 
« +1 
71 Vl i/ n -{- 1 yPi 
2 (2a - 3) 
n (* — 1) (/i — 2) \E 
cos (r +!);)( — 
1 
71 {71 — 1) \li 
/i-1 
ft .‘if/ ft\«‘ 
cos(«-2)x + YY - 
COS {/I - l)x 
cos ?ix 
ft\« 
E 
3 /ft\'^~“ 
+ iAi;l - 
a + 1 \E 
ft \“ 
(pj I cos(/i+ 2)xJ ( 11 a). 
n + 
These agree in giving at the surface of the rin 
V cos «X , (T \ cos (71 + 1) X cos (7 1 — 1) X 
— “Y r 771 m «- 1 
27rft/3,j 
4a |_ ??/ -4" 1 
2/1 
+ ih{(« +'i) o‘+ 2) + “ A - (--_7,(7-2) <=oU« - 2 ) x| + &c. (12). 
I have verified the above value of the potential at external points near the ring by 
evaluating the integral. The method here given is suggested by Laplace Mec. 
Cel.,’ Book 3, c. 6). It is not satisfactory, as it will not give the value of V further 
than the terms in cr". The form of the potential at points near the ring changes after 
these terms. Log E. is introduced, and it is found that some of the equations coincide 
so that there are not enough to determine the necessary constants. 
§ 9. [As illustrating this, and furnishiiig a result which will be used later, consider a 
distributio]! of matter of density a/^^ cos 2x on the surface of the ring. 
By § 22 of my previous paper, it is seen that 
Vq 
27ra“/32 
1 
2 
ft- 
E2 
cos 2x + 
2 
+ 2 
cos 3x “ cosxj 
i^) (2 “ 3 ^ 
3 
cos 4v ^ + &c. 
This agrees with (11a), except for the term 
4 loo— + 1 
G"^ It 
32 
The value of the potential, inside, is consequent!}^ given by 
Vi 
^TTCrfB^ 
•- 
ft- 
cos 
^X + 
4 
1 L® fi\ > 1 1’® 
1 -1 cos X + ^ ^ cos 3x 
ft” 
ft 
o 
cos 4x + 
cos 2x + 
2 
IB 
ft” 
2log 
+ &c. 
