Vibrations of a Dielectric Sphere. 441 



where p is of no order in tc~K Expanding in powers of cr, 

 jf m'Jm=n, 



i tan p \-cr sec 2 p = ip ±cr — ip z (l + n~ip/c~i)j { (k— 1)(1 4- n—ip/i 5 )^ 



— inp fc* — ip*/c~^} : 



rejecting cr 2 , and even cr in the last term on account of the 

 smallness of the denominators. This is justified by the final 

 result. I£ 



«=(jc-l)(l+n), /3 = p(^- / c-* + 7i/ci + p 2 /c-l) ; . (7) 

 then 



i (tan p — p) -f cr tan 2 p — — ?p 3 (l -f- ?i — ipfc~*)[(ct — ?/3) . 



But the same equation must be true for — p, so that 

 — i (tan p — p) -f cr tan 2 p = ?'/) 3 ( 1 -f ?i -{- ip/e~*)/(u -f z/3) ; 

 whence on addition 



2atanV = p'(/3* l -a/3,)/(" 2 + /3 2 ), ... (8) 



where a L = l-f??, /3 l = pK~K Writing now tan 2 p = p 2 , and 

 retaining only the most significant orders in k, 



a 2 + / 3 2 =(/,-l) 2 (l + ;0 2 + ^f a (l + ^)+«"V-l)} 2 



= /e 2 (lW/m) 2 , 

 /3xi — aj3i=(l-hn)p{fc* — K~*-TnfC* -\-p 2 tc~i) — p(l + n)(& — x~~) 



= (14- m'\m)(p*lK 4- m'\m)pK? 



(the rejection of lower orders was unsafe before), and therefore 



a=p 2 (m r + mp 2 jic)l(m + m')i& ) . . . (9) 

 from which 



X=±ipK-i+pXKm f + p 2 m)l{m + m f ) K \ . . (10) 



giving the formula already quoted when m' tends to zero. 



For the case treated in the earlier paper, when there is no 

 Newtonian mass, ??i = 0, so that 



\=±ip(C-* + p*lK? (11) 



Thus the vibration for the fixed sphere contains a factor 

 e -k x t w here Z: 1 = p 4 C/a/c 3 , and that for the charged and 

 movable sphere with no Newtonian mass contains e~ k - 1 where 

 k , 2 — p 2 0/aK 2 . 



It will be sufficient in applications of these results to 

 restrict attention mainly to the first vibration, for which 

 p = 4*493. 



Phil. Mag. S. G. Vol. 21. No. 124. April 1911. 2 G 



