Manchester Memoirs, Vol. xlii. (1898), No. 3. 5 

 increase the average density of the medium in the ratio 



I+iU. 



If we carry the approximation a stage further, we get 

 the dispersion-formula 



iV^=x., + i,W = iV^(.+1^5). (.0), 



where N^ denotes the refractive index for infinite wave- 

 length, as given by (19). 



2. We proceed to apply the preceding formulae to 

 the case of reflection. We will suppose that the string to 

 the left of the origin is unloaded, and that from the origin 

 onwards masses M are attached at equal intervals a, as 

 before** 



An incident wave 



in the unloaded portion may be regarded as giving rise to 

 a reflected wave 



^ = ^/^(^^+^) . . . (22), 

 and to a certain disturbance in the loaded medium. When 

 is real, and sin ka positive, this latter disturbance may 

 be represented by 



4=^/(^^/-rf) . . . (,3). 



and it is proposed to determine the coefficients of reflection 

 and transmission {^A and E). 



We will suppose that at the origin we have s^o. The 



kinematical condition to be satisfied at this point is 



ihA = B . . . (24). 

 The tension of the string immediately to the right of the 

 particle .f = o is 



* The case of sound-waves incident on a series of equidistant perforated 

 screens, as in the experiment described in Lord Rayleigh's Sound, § 343, is 

 mathematically equivalent. 



