18 • BELL SYSTEM TECHNICAL JOURNAL 



Where n is any integer positive or negative, including zero. If we take 



Ui = ^p -90 _ 



U2= d (5.11) 



C/g = + 90 



The matrices are consistent term by term. 



SECTION 6 

 Crystalline Dielectrics 



As a first application of the matrix algebra considered as a linear vector 

 function let us reconsider the problem of the crystal in an electric field. 



The relations of chapter II, equation (1) can be written in the abbreviated 

 form: 



D = DijE where D^ = D^r 



in accordance with the system of abbreviations adopted in the appendix. 

 If we put 



equation (1) can be written 



D = ^kr,E (6.1) 



In order to investigate the effects of crystal symmetry in determining 

 the least number of dielectric constants that are required for a given class of 

 symmetry it is desirable to find the electric induction D for any system of 

 axes. Suppose that we choose a system for axes .vi , .V2 , .vs related to .Vi , 

 Xi , Xz through the relations: 



Xi = fln.Ti + ai2.T2 + aisXs 



X2 = a2iXi + a22.T2 + 023-1^3 (6.2) 



X3 ^ asiXi + 032-^2 + ^33-^3 



where Cn is the cosine of the angle between .vj and .Vi , ai2 is the cosine of the 

 angle between .ti and X2 etc. 



Equation (6.2) can be abbreviated to 



x' = ax 

 where a is the matrix 



