102 BELL SYSTEM TECH NIC A L JOURNA L 



these has the property that it is unaltered by interchanging i and k and 

 therefore it is called a symmetrical tensor. The second has its components 

 reversed in sign when i and k are interchanged. It is therefore an antisym- 

 metrical tensor. Clearly in an antisymmetric tensor the leading diagonal 

 components will all be zero, i.e., those with i = k will be zero. Now since 



Wik= \ {wik + Wki) + h (u'ik — Wki) (70) 



we can consider any tensor of the second rank as the sum of a symmetrical 

 and an antisymmetrical tensor. Most tensors in the theory of elasticity 

 are symmetrical tensors. 



The operation of putting two suffixes in a tensor equal and adding the 

 terms is known as contraction of the tensor. It gives a tensor two ranks 

 lower than the original one. If for instance we contract the tensor ut Vk 

 we obtain 



UiVi = UiVi + U2V2 + U3V3 (71) 



which is the scalar product of u i and Vk and hence is a tensor of zero rank. 



We wish now to derive the formulae for tensor transformation to a new 

 set of axes. For a tensor of the first rank (a vector) this has been given 

 by equation (61). But the direction consines A to «3 can be expressed in 

 the form 



(72) 



(73) 



Similarly since a tensor of the second rank can be regarded as the product 

 of two vectors, it can be transformed according to the equation 



/ / /dXj \ /dXf \ dXj dXf .»,v 



\dXi / \dxis / dXi dXk 



which can also be expressed in the generalized form 



/ dXj dXf /-rv 



