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XIII. On the Vibrations of a Crystalline Medium. 

 By Haeold Hilton*. 



§ 1. TT is usual to picture a crystalline medium as 

 JL composed of similar " molecules " (or " particles," 

 not necessarily " molecules " in the chemical sense) at rest r 

 and arranged in a regular geometrical pattern f. Such 

 a picture is, however, exceedingly unsatisfactory, for the 

 molecules are certainly in motion. Moreover, in the usual 

 theories of crystal structure, no attention is paid to any 

 mechanical considerations. The difficulties inherent in 

 any theory which shows due respect for dynamical prin- 

 ciples are doubtless great. Here I attempt merely to- 

 give an indication of the kind of motions which the 

 molecules might be expected to make about their positions 

 of equilibrium : and with this object I discuss only a 

 simple case, namely, that of a holohedral orthorhombic 

 " crystal " whose Iran molecules have for their equilibrium 

 positions the points {pa, qb, re) referred to rectangular 

 Cartesian axes, where p can have any integral value from 1 

 to I, q any integral value from 1 to m, and r any integral 

 value from 1 to n ; while a, b, c are constants. These points 

 form the vortices of a " space-lattice " of the type T , in- 

 cluding T t and T r as special cases J. Consideration of the 

 orthorhombic symmetry shows that in the equilibrium 

 positions all the molecules are orientated in the same 

 way, with principal axes of inertia parallel to the axes 

 of reference. 



I suppose the molecules of this crystal to make small 

 vibrations about, their equilibrium positions. These vibra- 

 tions may be supposed known, if we can obtain (as we 

 shall) the " principal coordinates, "" also called " normal 

 modes," of the vibrating molecules, and the periods of 

 the principal oscillations. 



Various assumptions have been made about the nature 

 of the inter-molecular forces of a crystal. Some authors 

 assume that the appreciable influence of a molecule extends 

 to a distance which includes a very large number of other 

 molecules. Other authors assume that the influence of 

 each molecule is appreciable only on its. nearest neighbours 



* Communicated by the Author. 



t See the references on pp. 258, 259 of the author's ' Mathematical 

 Crystallography.' 



J See ' Mathematical Crystallography/ pp. 135-137. 



