360 Lord Rayleigh on the Propagation of Waves 



Accordingly by (10), 



n 2 = 16csin 2 (^ka) — 4csin 2 ka=16 c sin 4 (^ka), 

 or 



n = 4cism 2 (\ka) (21) 



Thus far we have considered the propagation of waves 

 along an unlimited series of bodies. If we suppose that the 

 total number is m and that they form a closed chain, ty must 

 be such that 



+,+,=*„ ....... (22) 



from which it follows that 



/3 = ka = 2s7r/m, . . (23) 



s being an integer. Thus (8) becomes 



n 2 =C -2C 1 cos (2s7r/m)-2C 2 cos (Isir/m)- (24) 



When the chain, composed of a limited series of bodies, 

 is open at the ends instead of closed, the general problem 

 becomes more complicated. A simple example is that treated 

 by Lagrange, of a stretched massless string, carrying a finite, 

 number of loads and fixed at its extremities *. The opsn chain 

 of m magnets, for which 



P = Mfi + f 2 ) 2 + K^ 2 + ^) 2 + ...+|(^- 1 + ^) 2 , . (25) 



is considered by Fitzgerald. The equations are 



ti + ^ 2 (2-n 2 ) + ^3 =0, 



t,-i + ^(2-n 2 )+^ r+1 =0, Y • • (26) 



"^m-2 + ^m-l(2 — U 2 ) + sjf m = 0, 



f m _ l+ f m (l-n 2 ) =0, 



of which the first and last may be brought under the same 

 form as the others if we introduce t/t and yfr m+l) such that 



fo + fi=0, f m +f m+1 =0. . . . (27) 

 If we assume 



f^ cos nt sin (r/3-±/3), .... (28) 



the first of equations (27) is satisfied. The second is also 

 satisfied provided that 



s'mmfi = 0, or fi = s7r/>n (29) 



* 'Theory of Sound,' § 120. 



