166 BELL SYSTEM TECHNICAL JOURNAL 



Similarly the third velocity v- = /3iF- = a- can also be used and equation 

 (21) reduces to 



"' + tA-, + -A-, = 0. (23) 



q2 — ^2 ^2 — ,y2 



This is a quadratic equation for the velocities v in terms of the principal 

 velocities a, b and c which are usually taken so that a > b > c. 

 Solving for the velocities, we obtain the quadratic equation 



V* - v''[nl(b^ + c2) + nlia'^ + c'~) + nl{a' + 6^)] 



2. ^ ' (24) 



+ nib~c~ + «2a'C" + Wsfl^^- = 0. 



Letting L = ni(b- — c~), M = ihic- — a?), N = nl(a- — b~) the solutions 

 for the velocities become 



Iv' = ni{b + c~) + Jioic" + a") + n^ia" + ft") 



/ . (25) 



This equation can be put into a simpler form if we change to the coordinate 

 system shown by Fig. 2. Here the rotated system is related to the original 

 system by three angles 9, (p, \f/. 6 is the angle between the Z axis and the 

 Z axis, <p is the angle the plane containing Z and Z makes with the X axis 

 while xj/ represents a rotation of the primed coordinate systems about the 

 Z axis. The direction cosines for the primed system with respect to the 

 normal system are designated by the matrix 



(26) 



where, in terms of 0, (f and \p, these direction cosines are, 



£i — cos 6 cos (p cos \p — sin (p sin \p, 



nil — cos 6 sin (p cos xp -\- cos (p sin \p, «i = — sin 6 cos xp 



ti = —cos 6 cos <p sin \p — sin (p cos \p, 



m-i = cos (p cos \p — sin v? sin xp cos 0, «2 = sin 6 sin i/' 



•^3 = cos (p sin ^, Mi = sin (^ sin 9, rh — cos ^. (27) 



If we take Z' as the direction of the wave normal, then in equation (25) 

 ill = k , ih = m-i , Hz = «3 



