366 Lord Rayleigh on the Dispersal of 



" Case 1. [From the general equation with conductivity 

 (C) zero and magnetic permeability (/jl) constant], 



(S + $)E-^4=0;. • • (D* 



or if, as before, k = 27r/\, 



* 



in which k is constant in each medium, but changes as we 

 pass from one medium to another. From (2) we see that 

 the problem now before us is analytically identical with 

 that treated in my book on Sound f, § 343, to which I 

 must refer for more detailed explanations. The incident 

 plane waves are represented by 



= e int {J (kr) + 2iJ 1 (kr) cos + . . . 



+ 2i m J m (kr) cos mO +...}; . . . (3) 



and we have to find for each value of m an internal motion 

 finite at the centre, and an external motion representing a 

 divergent wave, which shall in conjunction with (3) satisfy 

 at the surface of the cylinder (r = c) the condition that the 

 function (7i/K) and its differential coefficient with respect to 

 r shall be continuous. The divergent wave is expressed by 



Bo-^o + B Wr x cos 6 + B 2 f 2 cos 26 + . . . , . . (4) 



where yfr , ^r 1? &c. are the functions of kr defined in § 341. 

 The coefficients B are determined in accordance with 



B m f kcp^ J m (k'c)-k'c^ ml * ¥ - c JJUc) } 



= 2i m {lc'c J m (kc)J m '(V.6) -kc J m {k'c) JJ(kc)}, . . (5) 



except in the case of m = 0, when 2t m on the right-hand side 

 is to be replaced by i m %. In working out the result we 



* The numbering of the equations is changed, h is the component of 

 electric displacement parallel to z, K the specific inductive capacity, and 

 X the wave-length. 



t ' Theory of Sound,' vol. ii. _ Macmillan, 1st ed. 1878, 2nd ed. 1896. 



J Here k' relates to the cylindrical obstacle and h to the external 

 medium. 



