ELECTRICAL WAVE FILTERS 



451 



If we transform only the y and z axes, there results 

 h = h = nil = ni = 0, 

 h = 1, 

 fUi = fis = cos d, 

 — 712 = ni3 = sin 9. 

 Love ^^ gives the transformation for the stress and the strain func- 



(44) 



tions as 



Jy 



Xx 



= xj,^ + jym^ + z^n^ + y^mxn\, 



= Xxl^ + yymi + 22^2^ + yzm^Ui, 



= Xxh^ + y^Ws^ + z.nz^ + yzmsfis, 



= 2:j£;J2/3 + 2yym2m3 + 2z^n2nz + yzim^nz + m3«2), 



= Xx/i^ + Fj^wr + ^zWi^ + F22wiWi, 



= Xx^2^ + Fym2- + Z^7Z2" + F22W2W2, 



= Xxh-" + F,W32 + Z,«3- + F,2m3W3, 



= Xxhh + Yytn^nis + Z^n^Uz + Y^intons + m3W2). 



(45) 



(46) 



Substituting (44) in (45) and (46) and then expressing Xx, yy - • • 

 Xx, Yy . . ., etc., in terms of Xx\ yy' . . . Xx', Yy' . . ., etc., we 

 may substitute these values in equations (40) and (41) to give the stress- 

 strain and polarization in a crystal whose rectangular axes do not coin- 

 cide with the real optical and mechanical axis. Performing the above 

 operations, a new set of constants Si/, are obtained which are functions 

 of 6, namely: 



•^11 = Su, 



■^12' = |[5i2 + ^13 + (-^12 - ^13) cos 26 — Sii sin 20], 



•^13' = hL^iz + S12 + (5i3 - ^12) cos 26 + su sin 26~], 



Sn' = Sii cos 26 + (512 — ^13) sin 26, 



S22' = ^11 cos^ 6 -f 533 sin^ 6 -f 7su cos^ 6 sin 6 



+ (25i3 + Sii) sin^ 6 cos^ 6, 



S23' = 5i3(cos'' 6 + sin" 6) + Su{s'm^ 6 cos 6 - cos^ 6 sin 6) 



+ {su + .^33 — 544) sin^ 6 cos^ 6, 

 S2i' = — 5i4(cos" 9 — 3 sin^ 6 cos^ 6) + (25ii — 2^13 — Sa) cos^ 6 sin 



+ (2^13 — 2^33 + Sii) sin^ 9 cos 9, 



533' = -^33 cos'' 6 + 5ii sin"* 9 — 2sii sin^ 9 cos 



-f- (25i3 + -^44) sin- 6 cos^ 6, 



22 "The Mathematical Theory of Elasticity," Cambridge University Press, pp. 

 42 and 78. 



