1060 
MR. F. W. DYSOX OY THE POTENTIAL OF AN ANCHOR RING. 
giving on integration 
/3„ 
cr 
(2n + 1) ^,S;i+i 
'2n (n + 1) 
Therefore, adding the three integrals together, 
U — ^TT'^'a^c IL + ^ — 1 + ^ — (7 S 
n 
'2 L + 1 2» + 1 
O “1" 
2n {n + 1)_ 
AA + 4 (16) 
Let a-Q be the mean radius of the cross-section. 
Then 
7rao^ = TTCi~ il — 
(17). 
Let ^TrciQCQ be the volume of the ring. 
Then 
I 
n.? 
27r%o^Cn = 1 -f X ~ — cr S l^,Sn + \ • • 
\ 
Substituting, we obtain 
(18). 
8Cn 
U = \ log —" -f — S 
R 3 !L- ^ _ ^R R - 1) + 1) ' 
n 1 < T _j_ 
M. Poincare gives 
U = {log|" + i - 
(Tisserand, ‘ Mdc. Cel.,’ vol. 2, p. 166.) 
n 
[This is correct to the first power of cr. The term in cr"^ is important, for in 
the equilibrium position of the ring is of order cr®, and therefore this term is of 
the same order as the term in ^8/. By (10) and (13), the more correct value of U is 
{ L + i 
July 22, 1893.]* 
b i 3 3 (L II) . R ^ (J _5_\ 
cr cr P3 g (L 12) 
512 
_ Y 
A/ 
?i — 1 
% 
""A'A^+i 
{n - 1) (3?t + 1) 
471 {n + 1) 
]} . . (19). 
* Tills was inserted in consequence of an observation of one of tlie Referees, who noticed, what I had 
overlooked, that the less exact ralue of U failed in § 14 to give the correct equilibrium value of 
