STRAINED ELASTIC SOLIDS 



415 



orthogonal axes OX, OY, OZ, not necessarily coincident with 

 OX', OY', OZ'. Referred to these axes the coordinates of P" 

 are x, y, z and those of Q" are x -\- ^, y -{- tj, z -\- ^,so that the 

 local coordinates of Q" in a set of local axes through P" parallel 

 to OX, OY, OZ are ^, r/, f. The procedure now can be practi- 

 cally copied from the previous pages. Let us use a symbolism 

 similar to that employed above, and write 



dx dx 



«ii for — / ai2 for — ' 

 ox ay 



, dy 



a,, for -' 



etc. 



Then we find that 



^ = aii^' + aW + an^', ] 



V = «21^' + 0227?' + 023^', f 



r = 031^' + 03217' + flsar'- j 



(21) 



It follows that 



P"Q" - ail ' + a,r,'^ + ast' + 2airi'^' + 2a,t^' + 2a,^'n', (22) 

 where 



Oi = au^ + ^21^ + a3l^ 



a2 = ai2^ + «22^ + a32^ 



03 = ai3^ + 023^ + a33S . 



Gi = a.i2ai3 + 022^23 + «32a33, ( 



06 = aisfln + O23O21 + ^33031, 



de = CinCl'12 ~\~ ^21^22 4" a3ia32. 



(23) 



Now although an, a^, a^, etc., are not respectively the same as 

 Cii, ^12, 621 etc. (unless of course OX, OY, OZ should coincide with 

 OX', OY', OZ'), nevertheless a comparison of (8) and (22), 

 which are true for any values of ^', t] , ^' , shows that 



fli = ei, 02 = ei, az = €3, tti = Bi, 

 In consequence of (11), therefore, 



05 = 65, Oe = 66. 



Oi + 02 + 03 = ri^ + Ti^ + rs^, 

 a^as + 0301 + aitti — 04^ — 05^ — Oe^ 



r2V3^ + rsVi^ + riV2^ 



Ol 



Oe 



06 



06 

 02 



05 

 04 



03 



. (24) 



= r^T'^r^. 



