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BELL SYSTEM TECHNICAL JOURNAL 



control and the increase in the amount of power that may be controlled, 

 should result in a considerable simplification of future short wave 

 radio equipment. 



APPENDIX 



Elastic Equations 



The general elastic equations for any crystal are given below, X\ 

 Y' and Z' representing any orthogonal set of axes. 



- XJ = cii'xx + Cii'yy + ciz'z/ + ci 



— Yy = Ci2X/ + Cii'yy' + C23'z/ + C2 



— Zz = Ci3 Xx + C03 yy -\- C33 Zz + C3^ 



— Y/ = CuX:c' + Cii'yy + Cn'z/ + C44 



— ZJ = cx^lxj + ci'^yy + Ci^'zJ + C45 



— Xy = Ciq'xJ + C2/yy' + Csa'z/ + C46 



(3) 



When in quartz X', F' and Z' coincide with the crystallographic 

 axes of the material {X the electric axis, Y the mechanical axis, and 

 Z the optic axis), equation (3) reduces to equation (2) of the text. 

 In addition the following relations exist between the constants of 

 equation (2) because of conditions of symmetry 



Cl\ — C22, Cii — £"55, C66 — (cii - 



Cii = — C23 = C56. 



Cn)l2, 



'^is — C23 



The numerical values of these constants have been determined experi- 

 mentally by Voigt ^ and others. 



cii = 85.1 X 10 



C33 = 105.3 X 101 



10 y* /-.„ = A Qi; v inio "y- 



cm.^ 

 dy. 



cm.^ 



:cu = .6.95 X W 



cn = 14.1 X IQi 



Cii = 57.1 X W-^,cii = 16.8 X W 

 cm.'' 



r<-,6 = 39.1. 



cm.'' 



_dyi 



cm.^ 

 dy^ 

 cm.^ 



Using these constants it is possible to calculate the Ci/ for any orienta- 

 tion by means of transformation equations.^ The expressions giving 

 cu, C26.', ' • • Cqq (the constants relating to the Xy strain) in terms of 

 the Cij for rotation about the X axis, are given below, 6 being the 



«W. Voigt, "Lehrbuch der Kristallphysik," 1928, p. 754. 



^ A. E. H. Love, "Mathematical Theory of Elasticity," 4th ed., p. 43. 



