Vibrations of a Crystalline Medium. 151 



same material), but that the system is not fixed at or Z. 

 We readily find that in this case the expression for T 

 in § 2 (i.) still holds, but that the approximate expression 

 for U in § 2 (ii.) must be replaced by a quantity pro- 

 portional to 



(z,-z,y + (z,-z 2 ) 2 +... + (z n -z n . 1 y. . . (i.) 



Hence there are no lateral vibrations. 



The equations of § 2 (iii.) to determine the longitudinal 

 vibrations are replaced by 



z 1 =]i( — z 1 -\-z 2 ), z 2 =h(z 1 —2z 2 + z 3 ), ..., 



Z H -i — h(z n -2 — 2z n -i + Z n ), Zn = h(z n -1—Zn), • . (ii.) 



where h is some constant. 



Multiplying these equations (ii.) by 



cos^(2n — l)ky, cos^(2n — 3)ky, ..., cos \ 3&y, cos J £7, 



where 717 = 77-, and adding, we have 



&--4Asin 1 i*7.6, . . . ; . (iii.) 

 where 



f A = cos ^(2n — l)ky Zi + cos ^(2n — 3)kyz 2 + . . . 



+ COS±kyZ n (iv.) 



The periods of the principal oscillations are 



irh~% cosec 4^7, 



where 



k = 1, 2, ... , n — 1, and ny = it. 



We may suppose the centre of the mass of the system at 

 rest, which gives 



^ + ^+...4^ = 0; (v.) 



and, using this fact, equations (iv.) will give 



iiWr = oo_si(2fi-2r + l)7f 1 + oosi(2n-2r + l)27"f 1 +... 



+ cosi(2n-2r+l)(n-l)7S l -r • ( vi «) 



§4. Consider now the system of equal particles whose 

 equilibrium positions referred to rectangular Cartesian 

 axes are (pa, qb, 0), where p is any integer from 1 to I 

 and q is any integer from 1 to m, while a and b are 

 constants. 



Suppose that similar light springs of natural length a 

 join adjacent particles along each line parallel to the 



