Vibrations of a Crystalline Medium. 155 



will be replaced by equations of the type 



^p, 9, *" = " ,/ \?P—1, 9> r ^^P> 9> i* ~*~ *P+1> 9, *v 

 H" ff\Zp, 9-1, r i#, ?, »" ■" ^P. 9 + 1, ?y 



+ /i(2' p , (? , r _]-2^, (?jr + ^, (? ,, + i), . . (i.) 



unless /> = 1 or I, q=l or m, r=l or n. 



The quantities /, ^, h are constants whose magnitudes 

 depend on the unknown forces between adjacent molecules. 

 The terms in the last brackets on the right of (i.) are 

 replaced by 



( — ^,9.1 + ^,9,2) i£r=l, 

 and by (z Vy 9 , „_x — z p> q , ») if r = n ; 



and so on f or p and g. 



The (7 — l)(ra— l)(n — 1) principal coordinates £i,j,k of the 

 movement are now given by 



& JBt = X cos i(2l-2p+l)U x cos i (2m-2q + l)j!3 



x cos \ (2w — 2r + 1) £7 . z Pt q , r , 



where Icc = tt, mj3=ir, ny = 7r, while i is an integer between 

 1 and l—l,j an integer between 1 and m— 1, # an integer 

 between 1 and n— 1, p takes all integral values between 1 

 and Z, q takes all integral values 1 and m, and r takes 

 all integral values between 1 and ?i. 

 Then 



where 



X = 4(/ sin 2 ^iot + g sin 2 ^'/3 + h sin 2 ^ £7), 



the period of the corresponding oscillation being 2tt\~K 

 The analogy of the stretched string in § 2 suggests that 



the vibrations corresponding to small values of i,j, Jc are the 



most important in practice. 



The displacement of the centroid of those molecules of 



the crystal whose equilibrium positions lie in the line d' = p, 



y=q is u Pf q, where 



*- U p, q = Zp, 9, 1 "T" ~p, q, 2 ~T~ • • • *T &p, q, »• 



These centroids form a system oscillating in the manner 

 described in § 4 ; for equations (i.) give 



Up, q = f(u p -i, 2 — 2u p> 9 + ^+1, 7) +9( U P, 9-1 — -"+'• 5 + U P> ; + ') ' 



