156 Lord Rayleigh on the Maintenance of 



The third is 



{ a °-iY{( a >— ^-) 2 -l}=0, • • • (64) 



<2 3 2 



a 2 - 



a 3 



and so on. The equation (60) is thus equivalent to 



a,-— — .:.=±1; . . (65) 



and the successive approximations are 



N^+Di, N 2 =±D 2 , .... (Q6) 

 where 



D,'- IV' 



are the corresponding convergents to the infinite continued 

 fraction*. 



In terms of ® ? ®i> the second approximation to the equa- 

 tion discriminating the real and imaginary values of c is 



(eo-lXeo-9)-©^ ±6,(80-9). . . (67) 



One of the most interesting applications of the foregoing 

 analysis is to the case of a laminated medium in which the 

 mechanical properties are periodic functions of one of the 

 coordinates. I was led to the consideration of this problem 

 in connexion with the theory of the colours of thin plates. 

 It is known that old superficially decomposed glass presents 

 reflected tints much brighter, and transmitted tints much 

 purer, than any of which a single transparent film is capable. 

 The laminated structure was proved by Brewster ; and it is 

 easy to see how the effect may be produced by the occurrence 

 of nearly similar laminfe at nearly equal intervals. Perhaps 

 the simplest case of the kind that can be suggested is that of 

 a stretched string, periodically loaded, and propagating trans- 

 verse vibrations. We may imagine similar small loads to be 

 disposed at equal intervals. If, then, the wave-length of a 

 train of progressive waves be approximately equal to the 

 double interval between the loads, the partial reflexions from 

 the various loads will all concur in phase, and the result must 

 be a powerful aggregate reflexion, even though the effect of 

 an individual load may be insignificant. 



* The relations of determinants of this kind to continued fractions lias 

 been studied by Muir (Edinb. Troc. vol. viii.). 



