On the Propagation of Waves. 357 



P= . . . +iC ^ — Gityrfr-l— O^^V + l 



— 2 -v|r r ^ r _ 2 — Co-V^^r+2— .- • , .... (5) 



where A 1? A 2 , . . . Ci, C 2 , . . . are constants, finite for a certain 

 number of terms and then vanishing. The equation for 

 yp- r is accordingly 



A ^r r — Aif r _i- Ai^r+1 — A 2 T/r r _ 2 — A 2 i|r r+2 — . . . 



+ Co^r— Crfr-l — Cityr + l — C 2 ^ r _ 2 — C 2 ^ r+2 — . . . =0. (6) 



In the other equations of the system r is changed, but without 

 entailing any other alteration in (6). Since all the quantities 

 ■\fr are proportional to e int , the double differentiation is accounted 

 for by the introduction of the factor — n 2 . Making this 

 substitution and remembering that yfr r is also proportional to 

 e xr &, we get as the equivalent of any one of the equations (6) 

 ^(Ao-A^-A^-A^-^-A^- . . .) 



or 



2 _ C — 2C!Cos£a — 2C 2 cos 2&a — ., 



A —2A 1 <ioska—2A 2 cos2ka — ... , * ' 



in which /3 is replaced by its equivalent ka. By (7) n is 

 determined as a function of k and of the fundamental con- 

 stants of the system. 



In most of the examples which naturally suggest themselves 

 A 1? A 2 , . . . vanish, so that T has the same simple form as in 

 (1). If we suppose for brevity that A is unity, (7) becomes 



n^Co-SJOiCOs/ra— 2C 3 cos2#a — (8) 



When the waves are very long, k approximates to zero. 

 In the limit 



n 2 = C -2C 1 -2C 2 - (9) 



If we call the limiting value C, we may write (8) in the form 



n 2 = C + 4C 1 sin 2 (i*a)+4C 2 sin 2 (Aa) + (10) 



In an important class of cases C vanishes, that is the frequency 

 diminishes without limit as \ increases. If at the same time 

 but one of the constants Ci, C 2 , . . . be finite, the equation 

 simplifies. For example, if Cj alone be finite, 



n = 2Cfsin{±ka) (11) 



In any case when n is known V follows immediately. Thus 

 from (10) with C evanescent, we get 



. ., ka . «, Ska 



v , snr -r- . 2 , snr-a - 



£= c W +40 ^ +9Cs 73^ + (12) 



Phil. Mag. S. 5. Vol. 44. No. 269. Oct. 1897. 2 C 



