184 



BELL SYSTEM TECHNICAL JOURNAL 



(93) 



(94) 



Transforming the two terms mivi-i = Wri and Wnn = W21 by the tensor 

 transformation equation 



_ dxi dXj dXk dx( 



Wijh( ^~ ~ r r l^mnop 



CvvyTi OXfi (jXq OXn 



we find, for these two coefBcients, 



W12 = (wii + m-22 — 4w66) sin- 6 cos- 6 + 2(w62 — ^le) 



sin d cos' d + 2(w6i ~ Wie) sin ^ cos 6 -\- nin cos ^ + yrvn sin ^ 



W21 = (wii + W22 — 4w66) sin- ^ cos- + 2(wi6 — W62) 



sin B cos + 2(w26 — Wei) sin cos 6 -\- m^i cos + W12 sin ^ 



If W12 = W21 for all angles of rotation we must have 

 W16 + nhe = W61 + nieo 



For all the classes that W12 = nhi, either w^e = — Wie and m^o = — Wei or 

 else W16 = W26 = niei = m^o = 0. 



Now, if Z is a four-fold axis, as it is in the tetragonal and cubic systems, 

 then, for a 90° rotation, the value of nin or W21 must repeat. From the first 

 of (92) this means that 



W12 = W21 and nhi = niu 



For a trigonal or hexagonal system additional relations are obtained between 

 mm and mn , ^^22 and mn in the usual manner. Hence the photoelastic matrices 

 become, for the various crystal classes, 



(95) 



