STRAINED ELASTIC SOLIDS 403 



former position P'. This will bring the point Q" to R", where 

 Q"R" is parallel and equal to P"P'. The magnitude and 

 direction of the line Q'R" is the vector which, when estimated 

 for all Q' points in the neighborhood of P', would give us the 

 necessary information for calculating the strain. Now the 

 components of the vector length Q'R", the "differential dis- 

 placement" of Q' with reference to P', are ^" — ^', rj" — 7/', 

 ^" — f ' and are therefore equal to the expressions 



(en - l)r + e:2rj' + e^t',^ 



621^' + (622 - 1)^7' + e23r',[ (5) 



631^' + 63271' + (633 - l)i'',, 



which are linear functions of ^', t]' , f ' if en, 612, 613, ... 633 are 

 constants. 



H 



P' P 



Fig. 3 



Let us impose for a moment a simplifying condition with 

 regard to these nine strain constants and assume that 612 = 621, 

 623 = 632, esi = ei3. It will be very convenient for a moment to 

 write a for en — \,h for 622 — 1, c for 633 — 1, / for 623 or 632, g 

 for 631 or en, h for 612 or 621. Thus 



r' - r = ar + h' + gf',1 



r," -v' = H' + hr,' -^n',} (6) 



Taking P' as a local origin, and axes of reference through P' 

 parallel to OX', OY', OZ' ("local axes" at P'), let us suppose 

 the family of similar and similarly placed quadric surfaces con- 



