50 BELL SYSTEM TECHNICAL JOURNAL 



Multiplying (16.7) thru by the prefactor jc and cancelling the term in 

 jcU we have: 



jck-'j = ^ (16.9) 



This tells us that the velocity is a single valued function of the direction 

 of the current. 



With the idemfactor /, (16.7) may be written: 



lk~^ - — Ijj = n{k~^j)cn 



( -1 F' N~' 



If we multiply this thru by ( ^ ~ ~t ^ ] we get 



i = {k~^ - -^ ^) ' n{)r^3)on (16.10) 



Multiplying this thru by Uc and dropping the scalar factor {k" ]),n\ 



nik~^ -^l) w = (16.11) 



If the axes are so chosen that ^ is a diagonal matrix (16.9) and (16.11) 

 become: 



fji .2 .2 .2 



^ = |i + a + |i (16.12) j 



C- R\\ Rii ^33 



^' + ^^ + ^^ =0 (16.13) 



Examination of (16.13) shows that (16.11) must have two values of 

 F^ for each value of the vector normal n. As F is a single valued function 

 of y there must be two distinct values of 7 (/ and/' say) for any particular 

 n\ and given w, only waves having their current vectors in the directions 

 of / and /' can be propagated. A ray in the direction iV but not having 

 its j in one of the directions/ or/' will be broken up into two components 

 having their current vectors along / and /' respectively. 



If the velocity Fi corresponds to / and W to /' we have by means of 

 (16.10) since n' = n": 



j'cj" - n.(k^' -^iy'(k' - ^ ly n{k-Y)Mky)en 

 (The quantities in the braces are scalar) 



