MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
89 
since such integrals as 
vanish. Therefore 
1 sin 2(f) P„ (cos (f)) d(f) 
Jo 
2'rrah ^ 
V = - pj COS 2a 
R 
IH 
cos 2(f) P 2 + COS 2(f) ~ cos 2(f) P^, + 
+ ^-2 cos j - II cos Pi - ^ cos (/) Pi — . . . I d(f) 
5 S(f) Pg I d(f) 
+ “H“ 1^2 cos 3a i - o H 5 cos 3(f) F 
a cr 
2 R' 
. jr/r/) 
27r~a^c 
a- 
~ ' COS 2a 1 1 + A JTi + pG + • • • 
o o 
ZTr^'crc (T 
-A 
^ R 
2 cos “ P + B j;. + • . • 
(r 
_ 2ir(i?c a ^ f r , •< K " . 1 
“ R 2 ^ IP IP ■ f’ 
the integrals being found at once by using the expansions of P^ P 3 , &c., in cosines 
of multiples of (f). 
Substituting now — for cos a, ^ — 1 l*cr cos 2 a, ^ | for cos 3a : we find for the 
value of the potential 
op 
4P 3c „ 
a~ 
V = 2 «AA j _ I + (I „v + «*) i - f oV i - -/A- ^ + &c. I 
Similarly, the surface density a/3.^ cos 3y gives 
V = ^ - Ilf + . . . 
and the surface density cos 4y gives 
V = 2™'>oA 
MDCCCXCITI.—A. 
N 
