56 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Tlien 
Therefore 
p' = R2 _|_ COR xfj. 
p dp = sin xjj dxjj. 
1’lie voinine formed by tlie revolution of QTlli'Q' is 
'Zirp dp. LM. 
le potential of this at P 
'Http (/p.LAI 
” P 
= 'Itt 
— A:Trcrc 
(dv .sin yJr.dAjr'Ia sin -vp.cos a 
{R“ + cr — 2«R cos -^I /} 
silP.-vp dyjr 
— 2«i; cos-vp + a~} 
Therefore the poteni.ial of the whole ring 
M sin~ -xfr d-yjr 
TV Jo\/{P* — 2''d; cos \p +«"} ’ 
where M is the ma.ss of the rino’. 
Let there be two anchor rings of different radii, but of equal generating circles, 
having the same axis and centi’e. 
Let C and C' be the centres of the generating circles. 
Let P and P' be points on the axis such that CP = C'P'. 
If the densities of tlie rings be inversely jn’oportional to the distance of the centres 
of their generating (urcles from the axis of revolution, that is, if the densities be 
made such that the masses of the rings are equal, then the above formula shows that 
the potential of the C ilng at P is equal to the potential of the O' ring at P'. 
