98 Lord Kayleigh on the 



an infinitely small sphere is obtained from (29), (30), (31) by 

 writing h for h ; where by h is denoted the actual displace- 

 ment (parallel to z) within the particle, and by h the displace- 

 ment in the enveloping medium under the same electric force. 

 If K7 be the specific inductive capacity for the particle, the 

 ratio of h : h is 3K7 : K7 + 2K ; and in this ratio the results 

 expressed in (41), (42), (43) are to be increased. If we 

 extract the factors KAK -1 which there occur, we get 



K / + 2K^^^ ~K' + 2K\K' K/ ~ K' + 2K ' 



so that 



. 3(K'-K) ttT ay 

 /_ K' + 2K \*rr 26 ' (K0 ' ' ' ' ^V 



We learn from (62) that our former result as to the eva- 

 nescence of the secondary light along z is true for an infinitely 

 small spherical particle to all orders of AK. 



We will now return to the two-dimension problem with the 

 view of determining the disturbance resulting from the impact 

 of plane waves upon a cylindrical obstacle whose axis is 

 parallel to the plane of the waves. There are, as in the pro- 

 blem of reflection from plane surfaces, two principal cases — 

 (1) when the electric displacements are parallel to the axis of 

 the cylinder taken as axis of z, (2) when the electric displace- 

 ments are perpendicular to this direction. 



Case 1. — From (12), with C = 0, fju= constant, 

 / d 2 , cP \ 7t L 2 v h n 



or if, as before, k = 2ir/\, 



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

 pass from one medium to another. From (63) we see that 

 the problem now before us is analytically identical with that 

 treated in my book on Sound, § 343, to which I must refer 

 for more detailed explanations. The incident plane waves are 

 represented by 



pint pikx __ pint pikr cos 9 



= e int { J (kr) + 2tJi(*r) cos 6 + . . . + 2i m J m (kr) cos m0 + . . . } ; (64) 



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 (64) satisfy 

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



