Vibrations of a Crystalline Medium. 153 



OV being a line such that rotation through the angle e 

 (7r^e^0) about OV brings OX, OY, OZ to coincide with 



OA, OB, oc; 



The reader will readily prove that the direction-cosines 

 {hi h> ^3)5 ( m i> m 2> m s), { n i } noyfis) of OA, OB, 00 referred 

 to OX, OY, OZ are given by" 



Z 2 = 1 — 1(/> 2 — ^ 2 , m 1 = ^0(f)^ -yfrcos J*e, ?i 1 = l^ 4- (p cos ^e, 



&c. ; and that, when M] is considered fixed and M 2 moves 

 about which is also fixed, the component angular velocities 

 ©1, co 2 ) w s, of the molecule M 2 about the lines fixed in space 

 with which OA, OB, OC instantaneously coincide are 

 given by 



«i = 2 sec i 6 (^ + hQ + htfi + 1^), &c. 



Hence the kinetic energy o£ M 2 , when e and therefore 

 S, (f>) yjr are all small, is 



i(Afl 2 + B<£ 2 + (ty 2 ), 



where A, B, C are the moments of inertia of M 2 about its 

 principal axes OA, OB, OC. 



Suppose, now, that the coordinates of referred to the 

 principal axes of inertia of M x are (f, 77, c +■ f), where f, rj, £ 

 are small, so that M x and M 2 are adjacent molecules of the 

 crys'al along a line parallel to the axis of z. The work done 

 by the action of M] on M 2 . when M 2 is brought from the 

 position for which f, 97, f, 6, (/>, yfr are zero to its actual 

 position, is approximately a quadratic function of £, r\, f, 

 #, <£,^; the linear terms in these quantities being neglected, 

 since the total contribution of all similar linear quantities to 

 the work done in a small deformation of the crystal from a 

 position of equilibrium is zero. 



The orthorhombic symmetry of the medium shows that 

 this quadratic function cannot contain products of f, ?;, f, 

 0, <£, a/t, but must be a linear function of £ 2 , ?? 2 , f 2 , tf 2 , <fi' 2 , 1/r 2 . 

 For instance, the quadratic function must be unaltered if 

 f changes sign, and can therefore contain no term in 



fy], f;0, &C. 



Suppose, now, Mj and M 2 both receive small displacements 

 given respectively by the quantities fj, ?; l5 f 1? O u <pi, y x and 

 £2? V2) £2, #25 </>2? ^2- Then obviously 



f = ?2 — fl, V=V2 — Vli ?=?2 — Si 1 



and the reader will be able to show, using the parallelogram 



