152 Prof. H. Hilton on the 



x axis, and so for the y axis. If the particles make small vibra- 

 tions about their equilibrium positions, so that the particle 

 near {pa, qb, 0,) has coordinates {pa + x P}q , qb + y p , q , z p , q ), 

 then equations § 3 (ii.) are replaced by equations of the 

 form 



% g = f ( z p-h g ~ % z p, g + s p+i, g) + 9(. z p, q- i ~ % z p, q + z p.,q+i)i 00 



unless p = 1 or Z, <7=1 or m. 



The quantities /', g are constants. The terms in the last 

 brackets on the right of (1) are replaced by ( — z }>q + z 2 ,q) 

 if_p = l, and by (^-i, ? — z^g) if p = Z ; and so for g. 



The (/ — l)(m— 1) principal coordinates . (^ • are now 

 given by 



f. ,= X cos i(2Z - 2p + l)t« cos i(2m- 2q + l)j/3z p , q , 



p,q 



where /« = tt, ?n/3 = 7r, while i is an integer between 1 and 

 I — 1, y is an integer between 1 and m — 1, p takes all 

 integral values between 1 and I, and q takes all integral 

 values between 1 and m. 

 Then 



where \ = 4(/sin 2 \ioi-\-g sin 2 -^73), the period of the corre- 

 sponding oscillation being 27r\~2. 



A similar problem is that of the vibrations of the system 

 when the particles for which p = l and l, q = l and m, are 

 kept fixed. The reader will readiiy write down the solution 

 in this case, using § 2 to guide him instead of § 3, as is 

 done here. 



When a, b become small and p, q large, the motion of the 

 system approximates to that of a membrane. 



§ 5. We come now to the case of the crystal such as was 

 described in § 1. 



Suppose we have two adjacent molecules M l5 M 2 of the 

 crystal, similar in all respects. Let OA, OB, OC be the 

 principal axes of inertia of M 2 through its centroid O, and 

 let OX, OY, OZ be lines parallel to the principal axes 

 of Mi. The position of M 2 relative to M 1 is given by 

 six quantities, namely, the three coordinates of referred 

 the principal axes of M. 1 and the three quantities 



6 = 2 sin ±e cos VOX, 



= 2 sin Je cos VO Y, 

 -\/r = 2 sin -|e cos VOZ, 



