450 Mr. F. W. Dyson. [May 5, 



In this way the values of the integrals at the surface of the ring 

 are found and the required boundary conditions satisfied. 



3. The potential of an anchor ring at any point on its axis is 

 shown to be 



2M r sin*y^yr 

 7t J -/(R 2 — 2aRcos Y' + a 2 )' 



where R = CP and a =CA. 



This is reduced to elliptic functions and expanded in ascending 

 powers of a/R, giving 



LR 8 



Mfi_i*-±* Ac. 



8R 2 64 R 6 



, l 2 .3 2 ....(2tt-3) 2 (2n-l) a?« 1 

 2 2 .4 2 ....(2n-2) 2 (2?0 2 ( 2w + 2 ) R2B+1 J 



from which it is deduced that at any external point r,9 



y - ^0 



-v/0' 2 + c 2 — 2 cr sin cos 0) 



r 



\Z(r* + c 2 — 2 cr sin cos 0) 



8 ccZc Jo 



j 2a 2; * 1.3.5.. .. (2n-3) / _d \» f * <70 &j? 



2rc + 2 2 2 .4 2 .6 2 (2w) 2 \cdc) J n -/C^ + c 2 — 2 cr sin cos 0) ° 



This series of integrals is very convergent ; the first three are 

 reduced to elliptic functions, and the equipotential surfaces are drawn 

 for the ratios \, § , f , |, 1, of afc. 



4. The potential of a conductor in the form of an anchor ring is 

 shown to be of the form 



Aol + A^ 2 *L + A 2 a* I + &c. 



cac \cdcj 



Aq, A 1? A 2 , &c, are determined, and it is shown that on the snrface 

 of the ring 



T7_ A oJ\ ,„ 4X + 8\ + 5 ^ , 192X + 416X 2 + 448A + 179 A , 1 



v "71 0+2 ~t + 2 11 *-+... j> 



and 



dV A [ , /2\ + l 24A 2 + 7 2 \/o Z i \ 



+ ^- 3 .' cos 3 x + 1 i(L^ ff . cos 4x + &c . } , 



