U2 Mr. E. Talbot Paris on the Polarization of 



K may be replaced by m 2 , m being the refractive index of 

 the material composing the sphere relatively to the sur- 

 rounding medium, and then 7] —mif\. The functions -\fr n and 

 E„ are denned by 



^(,)=(-)>*1.3.5...(2« + l)(i|)"^, . (3 ) 

 E „ ( , )=( _)n.3.5...( 2 » + l)(i|p^. . i*) 



— LXylr n {7)) is the imaginary part of E„(?7). So that, sepa- 

 rating the real and imaginary parts, E n (^) = "^(77) — n/^177). 

 Tables of M* and i/r have been given by Rayleigh* for 

 arguments ranging from 1 to 3|. 



2. Numerical Results for a Perfect Conductor. 



If the sphere is supposed to be perfectly conducting the 

 expressions for M„ and N„ become much simpler, and the 

 numerical work is consequently much lighter. In order to 

 effect this calculation we have only to find the appropriate 

 values of M ?l and N„ and substitute them in equations (1) 

 and (2) given above. To this end we may make use of the 

 expressions given by Sir Joseph Thomson in ' Recent 

 Researches in Electricity and Magnetism,' page 446. These 

 expressions give the electric forces in the wave scattered by 

 a sphere having the character of a perfect conductor, but 

 they are applicable to ordinary conducting materials pro- 

 vided that the distance which the alternating currents (set 

 up by the incident vibration) penetrate into the sphere is 

 small compared with the diameter of the sphere. This dis- 

 tance is given by d=l^—^-\ •> where //, = the magnetic 



permeability, p==2irX the frequency of the incident wave, 

 and p= the specific resistance of the material composing the 

 sphere. There is no reason to suppose that this condition 

 could be fulfilled by any known metal when we are dealing 

 with oscillations of the frequency of light. Nevertheless, the 

 calculation is not without interest, for the above conditions 

 would be strictly fulfilled if we were dealing with Hertzian 

 waves and spheres (or spherical shalls) composed of a metal 

 such as copper. Moreover, the curves exhibiting the polari- 

 zation of the scattered light, obtained by assuming the sphere 

 to be perfectly conducting, do not differ very materially from 



* Proc. Roy. Soc, A, vol. lxxxiv. p. 39 (1910). 



