104 



BELL SYSTEM TECHNICAL JOURNAL 



, /dx'i^\ dx[ dx'i 



\ dxi I d.Ti 0x2 



+ ''P '-^ Tn + (g) 



8x2 dxi 



dxi dxi 

 dXi dxs 



dXi dxi _ dxi dxi 



i 22 -r T— - ^— i 23 — r — - — 1 k( 

 0X2 0X3 OXk dX( 



(79) 



5xi dxi dxi dxi ( dx 



~r -X — -z — i 31 "T -r — -7 — -/ 32 "rl -r- 

 d.T3 dxi dX3 dX2 \0iC3 



:)■ 



while the last equation takes the form 



/ _ dxi 8x2 . dxi dx2 „ , dxi 8x2 „ 



■t 12 — -^ — -z — i 11 ~r -7, — -;:— i 12 -r r — r — i 13 



dxi 0X1 dxi 6x2 oxi 0x3 



dxi dxo ™ , dxi 6x2 rp , dxi dxo ^ _„„,„... 



1 -7. — -z — i 21 "t" -r — - — -1 22 ~r r — - — i 23 — r — 'Z — i kf 



0X2 oXi 0X2 0X2 d.Vo 0X3 ax/c oXf 



, dxi 6x2 „ , dxi 6x2 „ , 

 ~r ~ — -;; — i 31 "T T— - -r — i 32 "T* 

 0X3 d.V] 0x3 0x2 



The general expression for any component then is 



r' . = ^^ f 

 '' dXk dxf 



dxi 8x2 

 dxf 



dxi dx'2 



dxs 6x3 



(80) 



(81) 



which is the transformation equation of a tensor of the second rank. Hence 

 the stress components satisfy the conditions for a second rank tensor. 

 The strain components 



•J XX '^xy "Jxz 

 •^yx '^yy '^yz 

 >J zx ^ zy ^ zz 



do not however satisfy the conditions for a second rank tensor. This is 

 shown by the transformation of strain components to a new set of axes, 

 which have been shown by Love to satisfy the equations 



Sxy — 2A^2'5'ii + 2viim2Syy + luirioSzz + (Aw2 + ^2Wi)5'j 



(82) 



+ (A"2 + fl\(2)S^z + (Wl"2. + m2lh)S:c 



