86 Lord Rayleigh on the 



further proof that much remains to be done before the elec- 

 trical theory of metallic reflection can be accepted as complete*. 



The same fundamental equations (8) and (9) will now be 

 applied to the problem of determining the effect on a train of 

 plane waves of a small variation in the quantities K and //, 

 which define the medium. A similar method will be adopted 

 to that already used for light in a paper " On the Scattering 

 of Light by small Particles "f, and in my book ' On the Theory 

 of Sound,' § 296, the principle of which consists in an ap- 

 proximation depending upon the neglect of the higher powers 

 of the small variations AK and A//,. 



Let us suppose that a train of plane waves, in which the 

 electric displacement is parallel to z, and magnetization parallel 

 to y, propagates itself parallel to x undisturbed until it falls 

 upon a region where the generally constant values of K and 

 \l become K + AK and fi + A/n. If AK and A//- were zero, 

 the wave would pass on as before ; but under the circum- 

 stances secondary waves are generated, which diverge from 

 the region of disturbance, and are ultimately, when AK and 

 Ayu, are small enough, proportional in magnitude to these 

 quantities. As the expression of the primary waves Ave may 

 take 



h z=e int e ik *, (23) 



and corresponding thereto, by (8), 



b = 4:7rhi- 1 K- 1 e int e ikx , .... (24) 



in which, if X denote the wave-length, & = 27r /X, and n/\ is 

 the velocity of propagation (K/jl)~*. The complete values of 

 the functions being represented, as before, by/, g, h, a, b, c, we 

 shall put 



/=/o+/i+/2 + - • -&c.> a=a + «i + . . . &c, 



/o . . . tt . . . being independent of AK and A/jl, /j . . . a x . . . 

 being of the first order, f 2 ...a 2 ... of the second order, and 

 so on, in these quantities. In the actual case f 0} g , a 0) c 

 vanish, and only h and b are finite. 

 From (8) and (9) with = 0, we get 



* July 15. — I see that Lorentz, in a pamphlet Over de Theorie der 

 Terugkaatsmg en Breking van net Licht (Arnhem, 1875), has developed a 

 theory of metallic reflection similar to that indicated in the text, and has 

 noticed the same difficulty in the application to experiment. 



t Phil. Mag. June 1871. 



