l^tuUi)np('d \"dii'alh)n.< m an Unliounded I dielectric. <>()!> 



sphere ami at all times : this result equally holds if the 

 vil)ration be not simple but compounded o£ a number oi 

 simple harmonic types. Now if we have two waves of the 

 same period travelling in opposite directions round the in- 

 tinitely thin ring the expressions for a, yS, y at a point P 

 [.I'ly yi, Ci) are of the type 



a = e'''^''j\nd/d(/i — md/dzi) {Cie'^^'-n + CV'''^"*"'*^^*"^^^-^^ 



where ?• is the distance of P from a point Q on the wire; 5 

 is the distance, measured along the ring, of Q from a fixed 

 point A at which the magnetic forces due to the two waves 

 are in the same phase; I, m, n are the direction cosines of the 

 tangent to the ring at Q; and the integral is taken all along 

 the ring. Let us take for the moment as axis of z the axis 

 of revolution of the ring, and as axes of x, v, radii of the 

 circular axis such that P lies in the plane i/z. Let the angles 

 AOX, XOQ, POZ be denoted respectively by cf), yjr, 0, and 

 the radius of the ring by a ; we have kci^^ct, where a is some 

 integer. The coordinates of P are then 0, R sin ^,R cos 6^ 

 and of Q (u;, y, ^), a cos >/'", a sin i/r, ; while /=— sin y\r^ 

 m= cos>/r, 11 = 0. Using Lord Kelvin^s symbol = to denote 

 equality to the order of approximation necessary we have 



>• = R — a sin ^ sin -v/r. 



•»:(«—'■) =^ -y«K(a^4-o</'-R + asin9sini/') 



1_ {;.-V''(*-'-)}:i=-_//c(?/i-3/)R-V''(*-»-) = -k sin 0.R-^^'-(^-'->, 



— {r- !(>«-(*-'•)}= -/A:(r^--)R-V''^*-^) = -i/ccos ^.R-^^'-^-''^ 

 azi 



Therefore, omitting the time factor^ the coefficient of Ci in x 

 =iVaR-^ cos ^.e'^'^^-K' I cos A^r . 6^^^'^+ «"^ e sin «/,) J^^ 



=<VR-i cos ^.e'^^^f-K' I COS ^p• cos <T{yfr-{- sin 6 sin ^)d^lr, 



=/(7R-i COS ^.^•'"«?-K^Fi(^), 



where FJ^) denotes the integral above. 

 Similarly the coefficient of C2 in a 



= iaR-^ cos e. e"^''-"^-^ 1 cos yjr cos o-( — -v/r + sin sin ^/r)(/x/r, 



= zaR-i cos e. e^^-'^^-'^^F^ie). 



Phil. Mag. S. G. Vol. 6. Xo. 36. dJec, 1903. 2 Y 



